# NCAR/TN-398 + STR NCAR TECHNICAL NOTE June 1995 

# A Description of the Fifth-Generation Penn State/NCAR Mesoscale Model (MM5)
Georg A. Grell1 

# Jimy Dudhia 2 

# David R. Stauffer3



# MESOSCALE AND

# MICROSCALE

# METEOROLOGY DIVISION 2 

# NATIONAL CENTER FOR ATMOSPHERIC RESEARCH BOULDER, COLORADO 1

# FORECAST SYSTEMS LABORATORY, NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION BOULDER, COLORADO 3 

# DEPARTMENT OF METEOROLOGY, THE PENNSYLVANIA STATE UNIVERSITY UNIVERSITY PARK, PENNSYLVANIATable of Con ten ts
Preface . . . . . . . . . . . . . . . . . . . . . . vii Ac kno wledgmen ts . . . . . . . . . . . . . . . . . ix Chapter 1 In troduction . . . . . . . . . . . . . . . . . . . . 1 Chapter 2 Go v

erning equations and n umerical algorithms . . 1 2.1 The h ydrostatic equations . . . . . . . . . . . . . . . 1 2.2 The nonh ydrostatic equations . . . . . . . . . . . . . 3 2.2.1 Complete Coriolis force option . . . . . . . . . . . . . 5 2.3 Nonh ydrostatic nite dierence algorithms . . . . . . . 6 2.4 h

ydrostatic nite dierence algorithms . . . . . . . . . 10 2.5 Time splitting . . . . . . . . . . . . . . . . . . . . 11 2.5.1 The nonh ydrostatic semi-implicit sc heme . . . . . . . . 11 2.5.2 The h ydrostatic split-explicit sc heme . . . . . . . . . . 13 2.6 Lateral boundary conditions . . . . . . . . . . . . . . 16 2.6.1 Sponge boundary conditions . . . . . . . . . . . . . . 16 2.6.2 Nudging boundary conditions . . . . . . . . . . . . . 16 2.6.3 Moisture v ariables . . . . . . . . . . . . . . . . . . 17 2.7 Upper radiativ e boundary condition . . . . . . . . . . 17 Chapter 3 The mesh-r enem en t sc heme . . . . . . . . . . . . 19 3.1 The monotone in terpolation routines . . . . . . . . . . 19 3.2 Ov erlapping and mo ving grids . . . . . . . . . . . . . 23 iii3.3 The feedbac k . . . . . . . . . . . . . . . . . . . . 23 3.3.1 A nine-poin t a v

erager . . . . . . . . . . . . . . . . 24 3.3.2 A smoother-desm oother . . . . . . . . . . . . . . . . 24 Chapter 4 Four-dimensional data assimilation . . . . . . . . . 25 4.1 Analysis n udging . . . . . . . . . . . . . . . . . . . 26 4.2 Observ ational n udging . . . . . . . . . . . . . . . . 32 Chapter 5 Ph ysical param eterizations . . . . . . . . . . . . . 37 5.1 Horizon tal diusion . . . . . . . . . . . . . . . . . . 37 5.2 Dry con v

ectiv e adjustmen t . . . . . . . . . . . . . . 37 5.3 Precipitation ph ysics . . . . . . . . . . . . . . . . . 38 5.3.1 Resolv able scale precipitation processes . . . . . . . . . 39 5.3.1.1 Explicit treatmen t of cloudw ater, rain w

ater, sno w, and ice . 39 5.3.1.2 Mixed-phase ice sc heme . . . . . . . . . . . . . . . . 45 5.3.2 Implicit cum ulus parameterization sc hemes . . . . . . . 46 5.3.2.1 The Kuo sc heme . . . . . . . . . . . . . . . . . . . 46 5.3.2.2 A modied Arak a

w

a-Sc h

ubert sc heme . . . . . . . . . . 50 5.3.2.3 The Grell sc heme . . . . . . . . . . . . . . . . . . . 64 5.3.3 P

arameterization of shallo w con v

ection . . . . . . . . . 70 5.4 Planetary boundary la y

er parameterizations . . . . . . . 73 5.4.1 Surface-energy equation . . . . . . . . . . . . . . . . 73 5.4.1.1 Net Radiativ e 
ux R

n . . . . . . . . . . . . . . . . . 74 5.4.1.2 Heat Flo w in to the Substrate H

m . . . . . . . . . . . . 78 5.4.1.3 Sensible-Heat Flux H

s

and Surface Moisture Flux E

s . . . 78 5.4.2 Bulk-aerodynamic parameterization . . . . . . . . . . 78 5.4.3 Blac k

adar high-resolution model . . . . . . . . . . . . 80 5.4.4 Vertical diusion . . . . . . . . . . . . . . . . . . . 84 5.4.5 Moist v ertical diusion . . . . . . . . . . . . . . . . 86
iv5.5 A

tmospheric radiation parameterization . . . . . . . . . 87 5.5.1 Longw a

v

e radiation . . . . . . . . . . . . . . . . . . 87 5.5.2 Short w

a

v

e radiativ e sc heme . . . . . . . . . . . . . . 90 Appendix 1 Glossary of sym bols . . . . . . . . . . . . . . . . 92 Appendix 2 Look-up table for transm issivit y . . . . . . . . . . 111 Appendix 3 Map Pro jections . . . . . . . . . . . . . . . . . . 112 Appendix 4 Land-Use categories . . . . . . . . . . . . . . . . 116 References . . . . . . . . . . . . . . . . . . . . . 117

vPREFA CE This tec hnical report describes the fth generation P enn State/NCAR Mesoscale Model Version 1, or MM5V1. It is in tended to pro vide scien tic and tec hnical documen tation of the model for users. Source code documen tation is a v

ailable as a separate Tec hnical Note (NCAR/TN-392) b y Haagenson et al. (1994). Comm en ts and suggestions for impro v

emen ts or corrections, are w elcome and should be sen t to the authors. (Last revision: June 1996)

viA CKNO WLEDGMENTS A large n um ber of scien tists, studen ts, and sponsors, too n umerous to men tion, ha v

e made signican t con tributions to w

ards dev eloping and impro ving this model. We w ould lik e to thank ev ery one who directly implemen ted c hanges and corrections in to the code, as w ell as ev ery one who helped to iden tify errors and problems. We w ould also lik e to extend our gratitude to the t w

o review ers (Prof. Tom Warner, and Dr. Jian-Wen Bao), whose commen ts impro v

ed the man uscr ipt signican tly . Computing for this model dev elopmen t w as undertak en on the CRAY YMP at NCAR, whic h is supported b y the National Science Foundation. The writing of this tec hnical documen t w as in part supported b y the Departmen t of Energy through gran t DEA105-90ER61070, and the California Air Resources Board through subcon tract A G/990-TS05-S02 from Alpine Geoph ysics.

ix1. In troduction This tec hnical report is a description of the fth-generation P enn State/NCAR Mesoscale Model (MM5). It is based on the original v ersion described b y An thes and Warner (1978). Although a few of the follo wing details of this model are w ell represen ted in An thes et al. (1987), extensiv e c hanges and increases in options ha v

e occurred. For completeness, those parts that ha v

e c hanged little or none will also be represen ted here. The documen t structure is as follo ws. In section 2 w e will describe the go v

erning equations, algorithms, and boundary conditions. This will include the nite dierence algorithms and time splitting tec hniques of both the h ydrostatic and the nonh ydrostatic equations of motion (h ydrostatic and nonh ydrostatic solv er). All subsequen t sections will describe features common to both solv ers. Section 3 will discuss the mesh-ren em en t sc heme, section 4 the four-dimensional data-assimilation tec hnique, and section 5 will focus on the v arious ph ysics options. 2. Go v

erning equations and n umerical algorithms 2.1 Hydrostatic model equations The v ertical 

-coordinate is dened in terms of pressure.

 = p p

t p s p

t

; where p

s

and p

t

are the surface and top pressures respectiv ely of the model, where p

t

is a constan t. The model equations are giv en b y the follo wing, where p



= p

s p

t

. Horizon tal momen tum ; @p 

u @ t = m 2



@p 

uu=m @x + @p 

vu=m @y  @p 

u

_  @ mp 



  @p  @x + @ @x  + p



f v + D

u (2 : 1

: 1) @p 

v @ t = m 2



@p 

uv=m @x + @p 

vv=m @y  @p 

v

_  @ mp 



  @p  @y + @ @y  p



f u + D

v (2 : 1

: 2) 1Temperature; @p 

T @ t = m 2



@p 

uT =m @x + @p 

vT =m @y  @p 

T _  @ + p

 ! c

p + p

 _ Q c p + D

T

; (2 : 1

: 3) where the D terms represen t the v ertical and horizon tal diusion terms and v ertical mixing due to the planetary boundary la y

er turbulence or dry con v

ectiv e adjustmen t. The heat capacit y for moist air at constan t pressure is giv en b y c

p = c

pd (1 + 0 : 8

q

v

), where q

v

is the mixing ratio for w ater v apor and c

pd is the heat capacit y for dry air. Surface pressure is computed from @p  @ t = m 2



@p 

u=m @x + @p 

v=m @y  @p 

_  @

; (2 : 1

: 4) whic h is used in its v ertically in tegrated form @p  @ t = m 2

Z 1 0



@p 

u=m @x + @p 

v=m @y 

d : (2 : 1

: 5) Then the v ertical v elocit y in 

-coordinates, _ 

, is computed from (2.1.4) b y v ertical in tegration. Th us _  = 1 p 

Z  0 

@p  @ t + m 2



@p 

u=m @x + @p 

v=m @y 



d 0 ; (2 : 1

: 6) where 

0

is a dumm y v ariable of in tegration and _  (

 = 0) = 0. In the thermodynam ic equation, (2.1.3), ! = dp dt and is calculated from

! = p



_  + 

dp  dt

; (2 : 1

: 7) where dp  dt = @p  @ t + m  u

@p  @x + v

@p  @y



: (2 : 1

: 8) The h ydrostatic equation is used to compute the geopoten tial heigh ts from the virtual temperature, T v : @ @ln (

 + p

t

=p 

) = RT v 

1 + q

c + q

r 1 + q

v

 1

; (2 : 1

: 9) 2where T v is giv en b y T v = T

(1 + 0 : 608 q

v

), and q

c

and q

r

are the mixing ratios of cloud w ater and rain w ater. 2.2 Nonh ydrostatic m od el equations For the nonh ydrostatic model w e dene a constan t reference state and perturbations from it, as follo ws: p

(

x; y; z; t ) = p

0

(

z

) + p

0 (

x; y; z; t )

; T (

x; y; z; t ) = T 0 (

z

) + T

0 (

x; y; z; t )

;  (

x; y; z; t ) = 

0

(

z

) + 

0 (

x; y; z; t )

: T ypically the temperature prole for the reference state ma y be an analytic function that ts the mean tropospheric temperature prole. The v ertical 

-coordinate is then dened en tirely from the reference pressure.

 = p

0 p

t p s p

t

; where p

s

and p

t

are the surface and top pressures respectiv ely of the reference state and are independen t of time. The total pressure at a grid poin t is therefore giv en b y p = p



 + p

t + p

0 ; where p



(

x; y ) = p

s

(

x; y ) p

t

. The three-dimensional pressure perturbation, p

0 , is a predicted quan tit y

. The model equations (Dudhia 1993) are then giv en b y the follo wing: Horizon tal momen tum ; @p 

u @ t = m 2



@p 

uu=m @x + @p 

vu=m @y  @p 

u

_  @ + uDIV mp   

@p 0 @x  p



@p  @x

@p 0 @

 + p



f v + D

u (2 : 2

: 1) @p 

v @ t = m 2



@p 

uv=m @x + @p 

vv=m @y  @p 

v

_  @ + vDIV mp   

@p 0 @y  p



@p  @y

@p 0 @

 p



f u + D

v (2 : 2

: 2) 3Vertical mom en tum ; @p 

w @ t = m 2



@p 

uw=m @x + @p 

vw=m @y  @p 

w _  @ + wDIV + p



g



0  

1 p 

@p 0 @ + T

0 v T T 0 p

0 T p 0

 p



g

[ (

q

c + q

r

)] + D

w (2 : 2

: 3) Pressure; @p 

p

0 @ t = m 2



@p 

up 0 =m @x + @p 

vp 0 =m @y  @p 

p

0

_  @ + p

0 DIV

m 2

p



p 

@u=m @x  mp



@p  @x

@u @ + @v=m @y  mp



@p  @y

@v @  + 

0

g
p @w @ + p





0

gw (2 : 2

: 4) Temperature; @p 

T @ t = m 2



@p 

uT =m @x + @p 

vT =m @y  @p 

T _  @ + T DIV + 1 c

p  p



Dp 0 Dt 

0

gp 

w D

p

0  + p

 _ Q c p + D

T

; (2 : 2

: 5) where DIV = m 2



@p 

u=m @x + @p 

v=m @y  + @p 

_  @

; (2 : 2

: 6) and _  = 

0

g p 

w m p

 @p  @x

u m p

 @p  @y

v: (2 : 2

: 7) The DIV terms are not in the h ydrostatic equations and arise because p

 is no w constan t in time. Th us the h ydrostatic con tin uit y equation no longer applies, lea ving the righ t hand side terms in (2.2.6) uncancelled b y the surface pressure tendency . The equations are th us in adv ectiv e form. Equation (2.2.4) can be deriv ed from the fully compressible mass con tin uit y relation and the perfect gas la w. The only term neglected in equations (2.2.1)-(2.2.5) is a diabatic term con tributing to the perturbation pressure tendency in (2.2.4). This term is negligible in normal meteorological regimes since it only forces a small div ergence (i.e. expansion) 4in regions of heating. The D

p

0

term in (2.2.5) is a small correction to D

T allo wing for horizon tal pressure dierences in thermal diusion. 2.2.1 Comp lete Coriolis force option In the nonh ydrostatic model it is possible to include the other componen ts of the Coriolis force that are neglected in the traditional appro ximation. The full Coriolis force leads to a small up w

ard/do wn w

ard acceleration on w esterly/easterly 
o ws and an w est w

ard/east w

ard acceleration on up w

ard/do wn w

ar d 
o ws in addition to the righ t

w

ard/left w

ard de
ection of horizon tal 
o ws in the northern/southern hemisphere. To determine this force, t w

o additional parameters are dened. We will refer to the other componen t of the Coriolis parameter as e

= 2
 cos 

, where 
 is the angular v elocit y of the earth and  is the latitude. The other new parameter is 

, whic h is the angular dierence bet w

een the y

-axis of the grid and true north. It is found from tan  = cos 

@=@y @=@y ; (2 : 2

: 8) where  is longitude. A special pro vision is made for the dateline. Th us 

is positiv e if north is rotated cloc kwise from the y

-axis. The momen tum equations are then giv en b y the follo wing: Horizon tal momen tum @p 

u @ t = m 2



@p 

uu=m @x + @p 

vu=m @y  @p 

u

_  @ + uDIV mp   

@p 0 @x  p



@p  @x

@p 0 @

 + p



f v p



ew cos  + D

u (2 : 2

: 9) @p 

v @ t = m 2



@p 

uv=m @x + @p 

vv=m @y  @p 

v

_  @ + vDIV mp   

@p 0 @y  p



@p  @y

@p 0 @

 p



f u + p



ew sin  + D

v (2 : 2

: 10) Vertical mom en tum ; @p 

w @ t = m 2



@p 

uw=m @x + @p 

vw=m @y  @p 

w _  @ + wDIV + p



g



0  

1 p 

@p 0 @ + T

0 v T T 0 p

0 T p 0

 p



g

[(

q

c + q

r

)] + p



e

(

u

cos  v

sin 

) + D

w

: (2 : 2

: 11) 52.3 Nonh ydrostatic Finite Dierence Algorithms The B-grid staggering of horizon tal v elocit y v ariables with respect to the other elds is sho wn in Fig. 2.1. Vertical v elocit y is staggered v ertically . Noting that the j index incremen ts in the x direction, and i

in the y direction, the con v

en tional notation will be as follo ws. a

x = ( a

i;j +

1 2 a

i;j 1 2

)

=

 x: (2 : 3

: 1)
a x = 1 2

(

a

i;j +

1 2 + a

i;j 1 2

)

; (2 : 3

: 2

a

) Multiple a v

eraging terms suc h as a

xyy can also be dened as successiv e a v

erages where the order of superscripts does not matter, e.g.,

a xyy = a

x

y

y

: Av eraging v ertically allo ws for non-uniform grid-lengths and nonlinearly v arying elds, suc h as temperature and w ater v apor, b y suitably w eigh ting the v alues. Th us for half-lev el elds a v

eraged to full lev els

a  = a

k

+

1 2

(



k 

k 1 2

) + a

k 1 2

(



k

+

1 2 

k

) ( 

k

+

1 2 

k 1 2

) ; (2 : 3

: 2

b

) while a v

eraging full-lev el elds to half lev els uses an equation similar to (2.3.2a). For temperature, a

is the poten tial temperature, and for w ater v apor, a

is log q

v

. The spatial dierencing of the terms in the horizon tal momen tum prediction equations is [including the map-scale factor m (

x; y )], @p  d

u @ t = m 2 2 4 u

x

p

 d

u m

xyy !

x + u

y

p

 d

v m

xyx !

y

3 5 ( p

 d

u



_ 

xy )

 + u

DIV xy mp  d  xy "p 0 x

y (

p 

)

x

y

p

0  p 

xy # + p

 d

f v + D ( p

 d

u

)

; (2 : 3

: 3)

6

# y(I)

x(J) I+1,J-1

I ,J-1

I-1,J-1 I+1, J I+1,J+1

I,J I ,J+1

I-1,J I-1,J+1 I,J-1 I,J

I-1,J-1 I-1,J (U,V)

Fig. 2.1 Horizontal grid structure in the model. 

# p* 

# (T qΦ σ  ̇ωqc

# qr

# R) ,,, ,, , , ,

7@p  d

v @ t = m 2 2 4 v

x

p

 d

u m

xyy !

x + v

y

p

 d

v m

xyx !

y

3 5 ( p

 d

v



_ 

xy )

 + v

DIV xy mp  d  xy "p 0 y

x (

p 

)

y

x

p

0  p 

xy #
p  d

f u + D ( p

 d

v

)

; (2 : 3

: 4) where p

 d

= p



xy , and DIV , the mass div ergence term, is giv en b y DIV = m 2 2 4 p

 d

u m

y

!

x + p

 d

v m

x

!

y

3 5 + p



_ 



: (2 : 3

: 5) The triple a v

eraging in the horizon tal momen tum adv ection terms follo ws that of the h ydrostatic model as discussed b y An thes (1972). The subgrid-scale and diusion operators are represen ted b y D (

a

) = K h

 x

2

(

a

xxxx + a

yyyy )+( K v

a

z

)

z

+ (PBL tendencies), where the fourth-order sc heme is modied to second-order near the boundaries. The coordinate v ertical v elocit y

, _  , is obtained from _  = 

0



g p  w m p



p



x x

u

xy m p



p



y y

v

xy ; (2 : 3

: 6) and the v ertical momen tum equation is @p 

w @ t = m 2

"



w

x

p



u m

y 

x + 

w

y

p



v m

x 

y

# ( p



w



_ 



)

 + w

DIV  + p



g



0 

 "

1 p 

p

0  1

p

0 T 0 p 0

T 

#
+ p



g



0 

 "

T

0 v T

 R c

p

p

0 T 0 p 0

T 

# p



g

(

q

c + q

r

)

 + D ( p



w

)

: (2 : 3

: 7) The pressure tendency equation, neglecting diabatic terms, is giv en b y @p 

p

0 @ t = m 2 "



p

0 x

p



u m

y



x + p 0 y

p



v m

x



y

# ( p



p

0 

_ 

)

 + p

0 DIV + p





0

g

w

 m 2

p



p 

 u m y



x (



p



x

)

x 1 mp



u



xy 8+  v m x



y (



p



y

)

y 1 mp



v



xy 

0

g m 2

p



w





; (2 : 3

: 8) and temperature tendency is dierenced as @p 

T @ t = m 2 "



T

x

p



u m

y



x + 

T

y

p



v m

x



y

# ( p



T



_ 

)

 + T DIV + 1 c

p  p



Dp 0 Dt 

0

g

p



w

 D ( p



p

0 )

 + p

 _ Q c p + D ( p



T

)

; (2 : 3

: 9) where Dp 0 =Dt is dierenced lik e the corresponding terms in (2.3.8). Moisture v ariables ha v

e similar adv ection forms to those in (2.3.8) and (2.3.9) except when using the upstream option where q

x

is replaced b y the upstream v alue alone.

92.4 Hydrostatic Finite Dierence Algorithms The h ydrostatic nite dierencing of adv ection, Coriolis and heating follo ws (2.3.3), (2.3.4) and (2.3.9) without the DIV terms. The pressure gradien t terms in (2.3.3) become P G = mp  d



x

y mR T v xy (1 + p

t

=p  d

)

p

 x

y

; (2 : 4

: 1) and lik ewise for the y-gradien t in (2.3.4). The surface pressure tendency is found from the in tegration o v

er all ( KMAX ) la y

ers of thic kness  (

k

), @p  @ t = m 2

KMAX X k =1 2 4 p

 d

u m

y

!

x + p

 d

v m

x

!

y

3 5  (

k

)

: (2 : 4

: 2) Then _  is found from do wn w

ard in tegration,
_  (

k

+ 1) = _ 

(

k

) @p  @ t

 (

k

) p  m 2 2 4 p

 d

u m

y

!

x + p

 d

v m

x

!

y

3 5  (

k

) p  ; (2 : 4

: 3) using the upper boundary condition that _  (

k = 1) = 0. The adiabatic term in (2.3.9), represen ted b y the second set of terms in square brac k

ets, becomes p



! in the h ydrostatic model, where ! is dened b y
! = dp dt = p



_ 

 +  

@p  @ t + m u

xy p

 x

x + m v

xy p

 y

y



: (2 : 4

: 4) The in tegration of the h ydrostatic equation to obtain geopoten tial heigh t, 

, in the h ydrostatic model is done as follo ws.  = R

T v L



ln (

 + p

t

=p 

)

; (2 : 4

: 5) where L = 

1 + q

c + q

r 1 + q

v

 1

; and allo ws for w ater loading when the explicit moisture sc heme is used. Because  is required on the v elocit y lev els (half-lev els), it has to be in tegrated rst bet w

een the surface, where  = 1 and  = gh (

h is the terrain heigh t abo v

e sea-lev el), and the lo w

est half-lev el using (2.4.5) with just the lo w

est-lev el v alues T v ; q v

; q c

; q r

. A t all other lev els (2.4.5) uses v ertical a v

eraging bet w

een t w

o lev els. 10The temporal dierencing in the h ydrostatic and nonh ydrostatic models consists of leapfrog steps with an Asselin lter. With this time lter, splitting of the solution often associated with the leapfrog sc heme is a v

oided. It is applied to all v ariables as ^ t = (1 2



) t + 

( t +1 + ^ t 1

)

; (2 : 4

: 6) where ^ is the ltered v ariable. The coecien t

 in the model is 0.1 for all v ariables. For stabilit y

, diusion terms are ev aluated on the v ariables at time t 1, as are the terms associated with the moist ph ysical processes. 2.5 Time splitting In both the nonh ydrostatic as w ell as the h ydrostatic n umerics, a time splitting sc heme is applied to increase eciency . Because the nonh ydrostatic equations abo v

e are fully compressible, they permit sound w a

v

es. These are fast and require a short time step for n umerical stabilit y

. F or the h ydrostatic equations, fast mo ving external gra vit y w a

v

es are the limiting factor. The tec hniques described next are designed to split these fast mo ving w a

v

es from the rest of the solution. 2.5.1 The nonh ydrostatic semi-im plicit sc heme For the nonh ydrostatic equations it is possible to separate terms directly in v

olv ed with acoustic w a

v

es from comparativ ely slo wly v arying terms, and to handle the former with shorter time steps while updating the slo w terms less frequen tly . The reduced equation set for the short time step mak es the model more ecien t. The separated equations only con tain in teractions bet w

een momen tum and pressure and can be written as: Horizon tal momen tum ; @u @ t + m  

@p 0 @x  p



@p  @x

@p 0 @

 = S

u (2 : 5

: 1

: 1) @v @ t + m  

@p 0 @y  p



@p  @y

@p 0 @

 = S

v (2 : 5

: 1

: 2) Vertical mom en tum ; @w @ t 

0  g p



@p 0 @ + g

p

0 p = S

w (2 : 5

: 1

: 3) Pressure; @p 0 @ t

+ m 2

p 

@u=m @x  mp



@p  @x

@u @ + @v=m @y  mp



@p  @y

@v @  11 0

g
p p

 @w @ 

0

gw = Sp 0

; (2 : 5

: 1

: 4) where the S terms con tain adv ection, diusion, buo y

ancy and Coriolis tendencies. These are k ept constan t during the sub-steps. Note that only part of the p

0 =p term is in (2.5.1.3), where the rest has been absorbed in the buo y

ancy term that con tributes to S

w

. The method of solution follo ws the semi-implicit sc heme of Klemp and Wilhelmson (1978) for the short time step. Starting with u; v; w; p 0

kno wn at time 

, rst the t w

o horizon tal momen tum equations are stepped forw ard to giv e

u



+1 and v



+1 whic h are then used in the pressure equation, giving a time-cen tered explicit treatmen t of horizon tally propagating sound w a

v

es. Vertical propagation of sound w a

v

es is treated implicitly b y making w



+1 and p

0 

+1 depend upon time-a v

eraged v alues of p

0

and w respectiv ely in (2.5.1.3) and (2.5.1.4). For instance, where p

0

appears in (2.5.1.3) it is represen ted b y p 0 = 1 2

(1 + ) p

0 

+1 + 1 2

(1 ) p

0 

; and similarly for w in (2.5.1.4). The parameter determines the time-w eigh ting, where zero giv es a time-cen tered a v

erage and positiv e v alues giv e a bias to w

ards the future time step that can be used for acoustic damping. In practice, v alues of = 0 : 2 0

: 4 are used. With second-order v ertical spatial deriv ativ es the nite dierence forms of equations (2.5.1.3) and (2.5.1.4) can be com bined, eliminating p

0 

+1 , in to a nite dierence equation for w



+1 , whic h is solv able b y direct recursion on a tri-diagonal matrix. The implicit v ertical dierencing sc heme allo ws the short time step to be independen t of the v ertical resolution of the model, whic h is importan t for eciency , and th us the step only depends upon the horizon tal grid length. Additionally , the div ergence damping tec hnique of Sk amaroc k and Klemp (1992) is used to con trol horizon tally propagating sound w a

v

es. This method is similar to using time-extrapolated pressure terms in (2.5.1.1) and (2.5.1.2), where in practice the extrapolation is about 0.1  

. T emperature and moisture are predicted using the normal leapfrog step,  t

, because they ha v

e no high-frequency terms con tributing to acoustic w a

v

es. The slo w terms for momen tum and pressure con tained in the S

-terms abo v

e are also ev aluated on these leapfrog steps, but for these v ariables the marc h from t 

t

to t

+ t

is split in to t ypically four steps of length   during whic h momen tum and pressure are con tin ually updated. 122.5.2 The h ydrostatic split-explicit sc heme When n umerically solving the h ydrostatic equations of motion, the stabilit y criterion is sev erely limited b y external gra vit y w a

v

es. These are v ery fast mo ving gra vit y w a

v

es that are small in amplitude (quasi-linear) and con tain only a small fraction of the total energy . Hence they c hange slo wly o v

er the time scale of the Rossb y w a

v

es. Because of this, splitting methods ha v

e been dev eloped to split these fast w a

v

es from the solution (similar also to the abo v

e method for the nonh ydrostatic equations to split sound-w a

v

es). From all the existing dieren t options, w e ha v

e c hosen a method dev eloped b y Madala (1981). This sc heme separates the terms go v

erning the gra vit y modes from those go v

erning the Rossb y modes. The term \split" here refers to the separation of the motion in terms of eigenmodes. Similar to the nonh ydrostatic method, the equations are rewritten in nite dierence form as @Ps u @ t + 

x

 = A

u

; (2 : 5

: 2

: 1) @Ps v @ t + 

y

 = A

v

; (2 : 5

: 2

: 2) @Ps T @ t + M2 

D = A

T

; (2 : 5

: 2

: 3) @Ps @ t + N

1



D = 0 ; and (2 : 5

: 2

: 4)  = M1 

T : (2 : 5

: 2

: 5) where the righ t hand sides c hange slo wly o v

er the time scale of the Rossb y-w a

v

es. Matrices M 1

,

M2 , and v ector N

1 are independen t of x

,

y

, and t

. Notice the similarit y to the nonh ydrostatic splitting method (equations 2.5.1-2.5.4). Ho w

ev er, rather then in tegrating the \fast" terms on a small time-step directly , the method described belo w only computes correction terms to the equations, making this process extremely ecien

t. To illustrate this, w e follo w Madala (1981). From the go v

erning equations he deriv es equations for the mass div ergence D and the generalized geopoten tial . They are @D @ t + [ 

2 x

+ 

2 y

] = 

x

A

u

+ 

y

A

v (2 : 5

: 2

: 6) and @

 @ t + M3 

D = M 1



A

T

: (2 : 5

: 2

: 7) 13In tegrating equations (2.5.2.1-2.5.2.3) from t 

t

to t

+  t

, where  t

is the time step of the slo w Rossb y modes, one gets p s

u

(

t

+  t

) p

s

u

(

t 

t

) + 2 t x

~  = 2 tA u

(

t

)

; (2 : 5

: 2

: 8) p s

v

(

t

+  t

) p

s

v

(

t 

t

) + 2 t

y

~  = 2 tA v

(

t

)

; (2 : 5

: 2

: 9) p s

T

(

t

+  t

) p

s

T

(

t 

t

) + 2 tM2 ~  = 2 tA T

(

t

)

; (2 : 5

: 2

: 10) where the operator ( ~ ) for the split-explicit sc heme is dened as
~ = 

  t m X n =1 (

t 

t

+ n





)

; where m = 

  t

. Denoting with superscript ex solutions computed using only the explicit time in tegration o v

er 2 t

, equations (2.5.2.8-2.5.2.10) can be written as p s

u

(

t

+  t

) + 2 t x

[

~  ( t

)] = p

s

u

ex (

t

+  t

)

; (2 : 5

: 2

: 11) p s

v

(

t

+  t

) + 2 t x

[

~  ( t

)] = p

s

v

ex (

t

+  t

)

; (2 : 5

: 2

: 12) p s

T

(

t

+  t

) + 2 tM2 [

~ D D (

t

)] = p

s

T

ex (

t

+  t

)

: (2 : 5

: 2

: 13) Here ( t

) and D (

t

) ha v

e been computed using the explicit time in tegration o v

er 2 t

. Similar, for the pressure tendency w e can write Ps (

t

+  t

) + 2 tN 1



[

~ D D (

t

)] = P

ex (

t

+  t

)

: (2 : 5

: 2

: 14) To nd equations for the correction terms on the left hand side of equations (2.5.2.11- 2.5.2.13), the div ergence and geopoten tial equations (2.5.2.6-2.5.2.7) are then solv ed o v

er the the small time-steps using [ D (

t

+ ( n + 1) 

) D (

t

)] [ D (

t

+ ( n 1) 

) D (

t

)] + 2 

(



2 x

+ 

2 y

)[( t

+ n





) ( t

)]

= 1 m

i

[ D

ex (

t

+  t

) D (

t 

t

)] (2 : 5

: 2

: 15)
14and [( t

+ ( n + 1) 

) ( t

)] [( t

+ ( n 1) 

) ( t

)] + 2 M3 [ D (

t

+ n





) D (

t

)]

= 1 m

i

[ ex (

t

+  t

) ( t 

t

)] : (2 : 5

: 2

: 16) The correction terms themselv es are in tegrated in equations (2.5.2.15)and (2.5.2.16), and then added to equations (2.5.2.11-2.5.2.14).  

, the timestep of the fast modes, of course v aries with the mode. For a clean separation of the modes, a v ertical normal mode initialization dev eloped and applied to the MM4/MM5 system b y Errico (1986) is used at the beginning of the model run to calculate the v ertical modes. In MM5, only the external and the fastest in ternal mode are being considered with dieren t time steps. This allo ws the time-steps of the slo w tendencies to be t wice as large as they w ere with the previously used Bro wn-Camp ana (1978) algorithm, and they are comparable to the ones used in the nonh ydrostatic n umerics.

152.6 Lateral Boundary conditions for the coarsest mes h doma in 2.6.1 Sponge Boundary Conditions The sponge boundary condition is giv en b y  @ @ t 

n

= w

(

n

)



@ @ t 

MC + (1 w

(

n

)) 

@ @ t 

LS ; (2 : 6

: 1) where n = 1 ;

2

;

3

;

4 for cross-poin t v ariables, n = 1 ;

2

;

3

;

4

;

5 for dot-poin t v ariables, represen ts an y v ariable, MC denotes the model calculated tendency ,

LS the large-scale tendency whic h is obtained either from observ ations or large-scale model simulations (onew a

y nesting), and n is the displacemen t in grid-poin ts from the nearest boundary ( n = 1 on the boundary). The w eigh ting coecien ts w

(

n

) for cross poin t v ariables (coun ting from the boundary poin ts in w

ard) are 0.0, 0.4, 0.7, and 0.9, while for dot-poin t v ariables they are equal to 0.0, 0.2, 0.55, 0.8, and 0.95. All other poin ts in the coarse domain ha v

e

w

(

n

) = 1. The abo v

e method cannot be used for the nonh ydrostatic part of the model. 2.6.2 Nudging Boundary Conditions The relaxation boundary condition in v

olv es \relaxing" or \nudging" the modelpredicted v ariables to w

ard a large-scale analysis. The method includes Newtonian and diusion terms  @ @ t 

n = F

(

n

)

F 1 ( LS MC ) F

(

n

)

F 2  2

( LS MC )

: n = 2 ;

3

;

4 (2 : 6

: 2) F decreases linearly from the lateral boundary , suc h that F (

n

) = 

5 n 3

 n = 2 ;

3

;

4

; (2 : 6

: 3) F (

n

)=0 n > 4

; (2 : 6

: 4) ; where F 1 and F 2 are giv en b y F 1 = 1 10 t (2 : 6

: 5) and F 2 =  s

2 50 t

: (2 : 6

: 6) This method is also used for the nonh ydrostatic part of the model to n udge the pressure perturbation to the observ ations or larger-scale model sim ulations. Ho w

ev er, for 16the nonh ydrostatic solv er the v ertical v elocit y is not n udged. It can v ary freely , except for the outermost ro ws and columns, where zero gradien t conditions are specied. For the v elocit y componen ts, the v alues at the in
o w poin ts are specied in a manner similar to the specication of temperature and pressure. The v alues at the out
o w boundaries are obtained b y extrapolation from the in terior poin ts. These boundary v alues are required only in the computation of the nonlinear horizon tal momen tum 
ux div ergence terms; They are not required in the computation of the horizon tal div ergence. 2.6.3 Moisture v ariables Cloud w ater, rain w ater, sno w, and ice are considered zero on in
o w and zero gradien t on out
o w. There is an option to specify the boundary v alues in the same w a

y as for the other v ariables (e.g., these v ariables ma y be kno wn in a one-w a

y nesting application). 2.7 Upper radiativ e boundary condition An option in the nonh ydrostatic model is the upper radiativ e boundary condition. Klemp and Durran (1983) and Bougeault (1983) ha v

e dev eloped an upper boundary condition that allo ws w a

v

e energy to pass through unre
ected. It can be expressed for h ydrostatic w a

v

es as ^ p

= N K ^ w; (2 : 7

: 1) where ^ p and ^ w are horizon tal F ourier componen ts of pressure and v ertical v elocit y respectiv ely ,

 and N are the densit y and buo y

ancy frequency near the model top, and K is the total horizon tal w a

v

en um ber of the Fourier componen t. This expression should be enforced for all componen ts if the energy transport is to be purely up w

ard with no re
ection. The upper boundary condition is com bined with the implicit pressure/v ertical momen tum calculation. Before either v alue at time n + 1 is kno wn, the v alues at the top model lev el ( w

1

is staggered half a grid length abo v

e

p

1

) can be expressed as p n

+1 1 = b + aw n

+1 1 ; (2 : 7

: 2) where the coecien t a

(

x; y; t ) is dependen t upon the thermodynam ic structure and the bottom boundary condition on w in the model column. It v aries within only 5 per cen t of a constan t v alue ev en with high terrain, and is also not strongly time-dependen t. The 17v alue of b

(

x; y; t ) depends on pressure and most of the pressure tendency terms, and both a and b

are kno wn at this stage. So transforming, assuming a

v

aries little about a non-zero constan t and taking a mean v alue a ^ p = ^ b + a

^ w: (2 : 7

: 3) Com bining (2.7.3) with the radiativ e condition (2.7.1) for w a

v

en um ber K = 2 = , taking N at the top of the model, and eliminating ^ p , giv es ^ w = K ^ b N

aK : (2 : 7

: 4) Using a limited-area 2D cosine transform, the forw ard transform, m ultiplication and bac kw ard transform can be com bined in to a single operator on the b

eld to giv e w

n

+1 1 . Hence w

IJ = I

+6 X i =

I 6 J

+6 X j =

J 6 ij

b

ij

; (2 : 7

: 5) where w e ha v

e localized the transform to 1313 poin ts, and arra y can be precalculated and k ept constan t for the time in tegration. The elemen ts of are found from ij = 6 X k =0 6 X l =0 

i



j



k



l 36

cos 2

ki 12 cos 2

lj 12 f

(

K )

; (2 : 7

: 6) with f

(

K ) = K N

aK and K = ( ^ k

2

+ ^ l 2

)

1 2

.

 = 1 except for limits of summ ations where  = 1 2

. Follo wing the suggestion of Klemp and Durran, the nite dierencing of pressure gradien ts and div ergences should be tak en in

to accoun t in dening the eectiv e w a

v

en um bers. F

or a B-grid staggering, the eectiv e w a

v

en um bers can be expressed in terms of the dimensionless w a

v

en um bers, k and l

, where ^ k = 2 

x sin k 12 cos l 12 ; (2 : 7

: 7

a

) ^ l = 2 

x sin l 12 cos k 12 ; (2 : 7

: 7

b

) and  x is the grid length. The sc heme is summ arized as follo ws; b y the precalculation of parameters a and N for the model domain, use of (2.7.6) to precalculate coecien ts , then implemen tation of (2.7.5) during the sim ulation. 183. The Mesh renemen t sc heme The 2-w a

y in teractiv e mesh renemen t sc heme is constructed to allo w for an arbitrary n um ber of o v

erlapping and translating rectangular grids with an arbitrary n um ber of renemen t lev els. The grids m ust be aligned with the model coordinates (no rotating meshes), and the mesh renemen t ratio of the temporal and spatial grid incremen ts is common for all meshes, and curren tly set to three. Vital parts of this renemen t sc heme are the in terpolation routines (Smolarkiewicz and Grell, 1992), whic h are used upon initialization of new nests as w ell as for dening the boundaries of the ne meshes. If the user can supply his o wn analysis for the ner grids (or a ner grid), the in terpolated elds can be o v

erwritten. In the follo wing section w e describe the heart of the sc heme, the monotone in terpolation routines. 3.1 The mono tone in terpolation routines The most vital elemen t of an y mesh renemen t sc heme is an accurate and ecien t in terpolation procedure. A complain t about traditional polynomial-tting methods used for in terpolating scalar elds dened on a discrete mesh is that they often lead to spurious n umerical oscillations in regions of steep gradien ts of the in terpolated v ariables. In order to suppress computational noise, whic h is c haracteristic of quadratic and higher-order in terpolation sc hemes, one often implemen ts a smoothing procedure or increased diusion. These, ho w

ev er, in troduce excessiv e n umerical diusion that smears out sharp features of in terpolated elds. A more adv anced approac h in v

ok es the so-called shape-preserving in terpolation, whic h incorporates appropriate constrain ts on the deriv ativ e estimates used in the in terpolation sc hemes (see Rasc h and Williamson (1990) for a review). In MM5 w e consider as an alternate approac h a class of sc hemes deriv ed from monotone adv ection algorithms (Smolarkiewicz and Grell, 1992). Smolarkiewicz and Grell (1992) sho w that the in terpolation problem becomes iden tical to the adv ection problem, when the distance v ector is replaced b y the v elocit y v ector (see also the end of this section). Here w e will describe the implemen tation of the adv ection algorithm used in MM5. The in terested reader is referred to Smolarkiewicz and Grell (1992) for a detailed deriv ation of the \adv ection-in terpolation" equiv alence. Since shape preserv ation and monotonicit y are importan t in the in terpolation problem, 19w e c hose the Flux Corrected Transport (F CT) sc heme that uses the high-order accurate constan t-grid-
ux dissipativ e algorithms dev eloped b y Trem bac k et al. (1987). We will rst describe, in abbreviated form, a general F CT algorithm, as used in MM5. Giv en the exactness of the alternate-direction represen tation of the in terpolation algorithm, it is sucien t to consider only one-dimensional F CT sc hemes.Starting with the onedimensional adv ection equation in 
ux-form @ @ t = @u @x ; (3 : 1

: 1) where  is a scalar v ariable adv ected with a 
o w eld u

(

x; t ), an F CT adv ection sc heme ma y be compactly written as  n

+1 i =  n

+1 i (

~ A

i +1 =

2 ~ A

i 1

=

2

)

; (3 : 1

: 2) where  denotes a lo w-order, monotone solution to (3.1.1), and ~ A

is the an tidiusiv e 
ux, limited suc h as to ensure that the solution (3.1.2) is free of local extrema absen t in the lo w-order solution. Note that ~ A i +1 =

2

= min 

1

; # i

; " i +1 A

i +1 =

2



+ + min 

1

; " i

; # i +1 A

i +1 =

2

 ; (3 : 1

: 3) where A

i +1 =

2

 F H i +1 =

2 F L i +1 =

2

; (3 : 1

: 4) with F H and F L denoting 
uxes from a high-order and a lo w-order adv ection sc heme, respectiv ely .[ ] +  max (0 ; ) and [ ]  min (0 ; ) are the positiv e- and the negativ e-part operators, respectiv ely , and " i  

MAX i 

n

+1 i A IN i + " ; # i

 

n

+1 i 

MIN i A OUT i + " ; (3 : 1

: 5

a; b ) where A

IN i ,

A

OUT i are the absolute v alues of the total incoming and outgoing A

-
uxes, (3.1.4), from the i

-th grid bo x, respectiv ely . "

is a small v alue, e.g.  10 15 , and allo ws for ecien t coding of -ratios when A

IN i or A

OUT i v

anish. The limiters 

MAX i and 

MIN i dene monotonicit y of the sc heme (i.e., b y design 

MIN i  

n

+1  

MAX i ), and they are traditionally specied (Zalesak 1979) as  MAX i = max 

n i 1

;  n i

;  n i +1 ;



n

+1 i

1

;



n

+1 i ;



n

+1 i

1

 (3 : 1

: 6

a

) 20 MIN i = min 

n i 1

;  n i

;  n i +1 ;



n

+1 i

1

;



n

+1 i ;



n

+1 i

1



: (3 : 1

: 6

b

) A shape-preserving in

terpolation sc

heme requires less restrictiv e monotonicit y constrain ts than a conserv ativ e adv ection sc heme. The minima o v

er ratios appearing in (3.1.3) ensure that the an tidiusiv e 
ux attributed to the i

+ 1 =

2 position on the grid does not con tribute to the generation of spurious extrema, either in gridbo x

i

or in gridbo x i + 1. Ho w

ev er, monotonicit y of the in terpolation sc heme only requires that 

n

+1 i = (

x

o

) is free of spurious extrema. Consequen tly , equation (3.1.3) ma y be replaced b y ~ A i +1 =

2

= min 

1

; # i

A

i +1 =

2



+ + min 

1

; " i

A

i +1 =

2

 : (3 : 1

: 3

0 ) Furthermor e, since the eectiv e 
o w eld is constan t, and therefore incompressible, the limiters in (3.1.6) ma y be simplied to  MAX i = max 

n i 1

;  n i

;  n i +1 ;



n

+1 i ;

 ; 

MIN i = min 

n i 1

;  n i

;  n i +1 ;



n

+1 i 

;

(3 : 1

: 6

0 a; b ) where the redundan t dependence of the limiters on  n

+1 i has been retained to ensure strictly nonnegativ e v alues of the ratios in (3.1.5) (cf., Section 3.1 in Smolarkiewicz and Grabo wski, 1990). Since the lo w-order solution ma y alw a

ys be written as an old v alue, min us the div ergence of 
uxes from the lo w-order sc heme, the en tire algorithm consisting of (3.1.2), (3.1.3 0 ), (3.1.4), (3.1.5), and (3.1.6 0 ) is in the form of a general adv ection sc heme. The adv ection sc hemes used to calculate the high- and lo w-order 
uxes for the abo v

e equations are from Trem bac k et al. (1987). They deriv e as follo ws. Starting with the 
ux form of the one-dimensional adv ection equation (3.1.1) in nite dierence form  n

+1 i = + 

n i

+ 

t  x

[ F i +1 =

2 F i 1

=

2

]

= 

n i

+  x  t "

X m b

m



n i +1+ m X m b

m



n i +

m

# ; (3 : 1

: 7) where F i +1 =

2

= X m b

m



n i +1+ m (3 : 1

: 8) and F i 1

=

2

= X m b

m



n i +

m (3 : 1

: 9) 21w ere used. F

ollo wing Trem bac k et al. (1987), the solutions for the ev en-order sc hemes whic h are used in the mesh renemen t sc heme are then giv en b y F i +1 =

2



t  x = + 2

( 

i 

i +1 )+ + 2 2

2( 

i

+ 

i +1 ) (3 : 1

: 10) for second order accuracy; F i +1 =

2



t  x = + 12 (



i 1 7



i 7



i +1 + 

i +2 ) + 2 24

(



i 1 15 

i

+ 15 

i +1 

i +2 ) + 3 12

( 

i 1

+ 

i

+ 

i +1 

i +2 ) + 4 24

( 

i 1

+ 3 

i 3



i +1 + 

i +2 ) (3 : 1

: 11)
and for fourth order accuracy; F i +1 =

2



t  x = + 60 ( 

i 2

+ 8 

i 1 37 

i 37 

i +1 + 8 

i +2 

i +3 ) + 2 360

( 2



i 2

+ 25 

i 1 245 

i

+ 245 

i +1 25 

i +2 + 2 

i +3 ) + 3 48

(



i 2 7



i 1

+ 6 

i

+ 6 

i +1 7



i +2 + 

i +3 ) + 4 144

(



i 2 11 

i 1

+ 28 

i 28 

i +1 + 11 

i +2 

i +3 ) + 5 240

( 

i 2

+ 3 

i 1 2



i 2



i +1 + 3 

i +2 

i +3 ) + 6 720

( 

i 2

+ 5 

i 1 10 

i

+ 10 

i +1 5



i +2 + 

i +3 ) ; (3 : 1

: 12)

for sixth order accuracy; is dened as

= U 

t  x

: (3 : 1

: 13) In MM5, equations (3.1.10 - 3.1.12) are used together with (3.1.1), (3 : 1

: 3

0 ), (3.1.4), 3.1.5), and (3 : 1

: 6

0 ) to solv e the in terpolation problem. Note that the v elocit y v ector, is replaced b y the distance v ector, X d

, whic h, with a mesh-renem en t ratio of three, simply becomes 1 =

3 or 2 =

3. For in terpolating boundary conditions to the ner meshes, fourth order accuracy 22is used, while for new nest initialization, sixth order accuracy is used. While the new nest initialization co v

ers the whole domain, boundary in terpolation is performed for the outermost 2 ro ws and columns of the nest. Tw o ro ws w ere necessary to ensure that the same operators w ere applied to eac h nested grid-poin t (including fourth-order diusion). 3.2 Ov erlapping and mo ving grids The mesh-renem en t sc heme allo ws for o v

erlapping grids on the same nest-lev el. To ensure n umerical stabilit y

, the solution in the o v

erlap region has to be iden tical. To accomplish this, after eac h time-step of the grids in question, the boundary conditions in the o v

erlap regions are pro vided b y the o v

erlapping mesh. It is v ery importan t that this procedure be performed at ev ery timestep. Nests can also be mo v

ed at an y time in the forecast. This can be done man y times, and for an y distance (in teger n um ber of grid poin ts). The user ma y also mo v

e the nests automatically if he supplies an algorithm to do so. Upon a mo v

e, a new nest initialization is performed rst. Then all high-resolution information from the previous location of the mesh is used to o v

erwrite the elds of the newly initialized mesh. Therefore, to ensure best use of high resolution information, it is better to mo v

e a nest more often and for a smaller distance. 3.3 The feedbac k to the coarser grids Since the mesh renemen t ratio in MM5 is set to three, a higher resolution mesh has to be in tegrated three times as often as its \Mother Domain"(MD), where MD means the coarser domain from whic h it gets its boundary conditions. To k eep the solutions in a 2-w a

y in teraction from div erging, the meteorological elds ha v

e to be fed back from the higherresolution mesh to its MD. This is done at the end of the three time-step in tegration. Naturally , when this is done without smoothing or a v

eraging, the solution on the MD will appear somewhat noisy (diluted with small-scale information). To a v

oid n umerical instabilit y

, the follo wing methods are supported in MM5 to remo v

e non-resolv able noise from the MD. Note that these smoothers are only applied o v

er an in terior area that is completely determined b y the higher resolution domain. It is importan t that input in to the nest, and feedbac k bac k to the MD does not o v

erlap. The smoother that is used b y 23the MM5 system in v arious forms w as dened b y Shapiro (1970) as
 (

i; j ) = (

i; j ) +  2

(1 

)( (

i

+ 1 ; j ) + (

i 1

; j ) + (

i; j + 1) + (

i; j 1) + 4 (

i; j )) + 

2 4

( (

i

+ 1 ; j + 1) + (

i

+ 1 ; j 1) + (

i 1

; j + 1) + (

i 1

; j 1) 4 (

i; j )) (3 : 3

: 1) 3.3.1 A Nine-poin t av erager This method w as in the original MM4 nested v ersion (Zhang et al. 1986). It is a feedbac k method that a v

erages information for a whole MD grid bo x (surrounding and cen tering on the nested grid poin t). Ho w

ev er, it does not tak e out all non-resolv able information on the MD. It also imposes a sev ere restriction on the terrain for the h ydrostatic model. In case of o v

erlapping and mo ving nests on sev eral nest lev els, it is v ery elaborate and complicated to apply . It is still an option in the model, because it ma y be useful for simpler applications (lik e one coarse and one nested domain). Ho w

ev er, care m ust be tak en to create a terrain data set that is consisten t with this method. The operator that is applied to the nested grid-poin ts (note that nothing is done to the MD) is dened b y using  = 0 : 5 in (3.3.1). 3.3.2 A Smoother-Desm oother The smoother-desm oother is a lter that remo v

es 2 x w

a

v

es and damps short w a

v

es, but lea v

es long w a

v

es almost unaected. It is m uc h more selectiv e than diusiv e smoothers. It is applied to the \coarser grid" only in the area where the coarse grid v alues are o v

erwritten with the nested grid v alues. A single pass of the smoother-desm oother in v

olv es t w

o steps. Equation (3.3.1) is used rst to smooth the elds, then to desmooth the elds. 

1 = 0 : 50 is used for the smoothing coecien t, and 

2 = 0

: 52 for the desmoothing coecien t. The rst step strongly smoothes the eld, completely remo ving the 2 x w

a

v

e, and the second step attempts to restore the other w a

v

es to their original amplitudes. There are t w

o passes of the smoother-desm oother applied in the model.

244. F our Dimensional Data Assimilation (FDD A) The concept of com bining curren t and past data in an explicit dynamical model suc h that the model's prognostic equations pro vide time con tin uit y and dynamic coupling among the v arious elds has become kno wn as four-dimensional data assimilation (FDD A). Curren t in terest in the use of FDD A in mesoscale models, for either model initialization (dynamic initialization) or for use of the model as an analysis/researc h tool (dynamic analysis), is a logical extension of the traditional link bet w

een ob jectiv e analysis methods and dynamic relationships. Curren tly , t w

o ma jor t ypes of FDD A are used operationally and in researc h. The rst is an in termitten t process of initializing an explicit prediction model, using the subsequen t forecast (t ypically 3-12 h) as a rst guess in a static three-dimensional ob jectiv e analysis step, and then repeating the process for another forecast cycle. The second is a con tin uous (ev ery model time step) dynamical assimilation where forcing functions are added to the go v

erning model equations to gradually \n udge" the model state to w

ard the observ ations. This con tin uous n udging t ype of FDD A is used in the PSU/NCAR modeling system. Nudging w as rst dev eloped and tested at P enn State b y Kistler (1974), under Prof. J. Ho v

ermale, and b y An thes (1974), and Hok e and An thes (1976). See Stauer and Seaman (1990) for an historical o v

erview of the tec hnique. Nudging or Newtonian relaxation is a relativ ely simple but v ery 
exible tec hnique: the data used for n udging can be of an y t ype, measured or deriv ed, analyzed to a grid for assimilation in to the model or inserted as individual observ ations. Gridded analyses of the observ ations that are assimilated can be obtained b y successiv e correction, v ariational, or statistical optimal in terpolation (OI) tec hniques, and the w eigh ts used when n udging to individual observ ations can be simple Cressman- t

ype (distance-w eigh ted) functions or more complicated statistical functions based on OI. It can be sho wn that successiv e corrections, OI, and v ariational approac hes are all fundamen tally related to the \idealized analysis" and th us to eac h other. In fact, n udging is basically a successiv e-correction tec hnique whic h uses a n umerical model to include the time dimension. The method of Newtonian relaxation or n udging relaxes the model state to w

ard the observ ed state b y adding, to one or more of the prognostic equations, articial 25tendency terms based on the dierence bet w

een the t w

o states. The model solution can be n udged to w

ard either gridded analyses or individual observ ations during a period of time surrounding the observ ations. These t w

o tec hniques, hereafter referred to as \analysis n udging" and \obs n udging", respectiv ely , can be used individually or sim ultaneously on an y MM5 model grid. F

or guidance in selecting whic h n udging tec hnique(s) to use for y our particular application, as w ell as suggested parameter specications, see Stauer and Seaman (1990), Stauer et al. (1991) and Stauer and Seaman (1993). 4.1 Analysis Nudging The analysis-n udging term for a giv en v ariable is proportional to the dierence bet w

een the model sim ulation and an analysis of observ ations calculated at ev ery grid poin t. The general form for the predictiv e equation of v ariable (

x

; t

) is written in 
ux form as @p  @ t = F

(

; x

; t

)+ G 

W (

x

; t

)



 (

x

)

 p



( ^ 0 )+ G

p





W p 





p



(

x

)

 (^ p  0 p



) (4 : 1

: 1) All of the model's ph ysical forcing terms (adv ection, Coriolis eects, etc.) are represen ted b y F

, where are the model's dependen t v ariables, x are the independen t spatial v ariables, and t is time. The second and third terms on the righ t of (4.1.1) are similar in form and represen t the n udging terms for and p*, respectiv ely . Due to the 
ux form of the predictiv e equation, the third term appears in (4.1.1) when n udging pressure in the con tin uit y equation of the h ydrostatic v ersion of MM5. (Note that this term is zero in the nonh ydrostatic v ersion of MM5 because p



is computed from the h ydrostatic reference state and is constan t in time.) With G

p



= 0, or in the nonh ydrostatic v ersion of MM5, (4.1.1) simplies to @p  @ t = F

(

; x

; t

) + G 

W 

 (

x

)



p



( ^ 0 ) (4 : 1

: 2) The n udging factor G determines the magnitude of the term relativ e to all the other model processes in F

. Its spatial and temporal v ariation is determined mostly b y the four-dimensional w eigh ting function, W , whic h species the horizon tal, v ertical and time w eigh ting applied to the analysis, where W = w

xy w



w

t

. The analysis qualit y factor, 

, whic h ranges bet w

een 0 and 1, is based on the qualit y and distribution of the data used to 26produce the gridded analysis. The estimate of the observ ation for analyzed to the grid is ^ 0

. The n udging factor G is dened based on scaling argumen ts. Because the n udging con tribution is articial, it m ust not be a dominan t term in the go v

erning equations and should be scaled b y the slo w

est ph ysical adjustmen t process in the model (inertial eects). Th us the G is usually dened to be similar in magnitude to the Coriolis parameter, and it m ust also satisfy the n umerical stabilit y criterion, G  1 

t

. T ypical v alues of G are 10 4

s 1 to 10 3

s 1 for meteorological systems, where v alues of G = 310 4

s 1 to 610 4

s 1

are usually \large enough". A v alue of G whic h is too large will force the model state too strongly to w

ard the observ ations. This is undesirable because (a) the abilit y of the model equations to resolv e mass-m om en tum im balances will be decreased; and (b) the abilit y of the model to generate its o wn mesoscale meteorological structures (e.g. fron ts, squall lines) will be impaired b y hea vy insertion of the observ ed analyses. Suc h problems arise because the analyses ma y not resolv e these mesoscale structures or ma y be con taminated b y observ ational and analysis errors. On the other hand, if G is too small, the observ ations will ha v

e minimal eect on the ev olution of the model state, allo wing phase and amplitude errors to gro w. For simplicit y

, if w e drop the ph ysical forcing terms F from (4.1.2), and assume W (

x

; t

)=1 ;

@p  @t

= 0 and the observ ational analysis is perfect and time in v

arian t, then @ @ t = G ( ^ 0 ) (4 : 1

: 3) whic h has the solution = ^ 0

+ ( i ^

0

)

e G t (4 : 1

: 4) where i

is the initial v alue of at the start of the n udging period. Therefore, the model state approac hes the observ ed state exponen tially with an e-folding time of T G = 1 G , whic h is about 0 : 93 h for G = 310 4

s 1

. This implies that v ery high frequency uctuations in the data, as migh t be a v

ailable from wind prolers or Doppler radars (sa y

, ev ery 5 min), w ould not be retained w ell unless G w

ere m uc h greater; but then the n udging term ma y not be small compared to some terms of F. 27Nudging the v orticit y is a alternativ e method whereb y the model's div ergen t wind is allo w

ed to freely respond in the model's geostrophic adjustmen t process. Equation 4.1.3 can be modied for u and v suc h that @u @ t = G

u

 @ @y (

^  0 

) (4 : 1

: 5) @v @ t = G

v

 @ @x (

^  0 

)

; (4 : 1

: 6) where 

is the model v orticit y and ^  0

is the analyzed observ ed v orticit y

. Letting a constan t G = G

u

= G

v

and forming the v orticit y equation from (4.1.5) and (4.1.6) w e get @ @ t = G

r 2

(

^  0 

) (4 : 1

: 7) Therefore, the model v orticit y is diused to w

ard the observ ed v orticit y

. Ho w

ev er, the Laplacian in (4.1.7) in troduces a scale dependence when n udging v orticit y

. The model v orticit y will be diused more strongly to small-scale features in the observ ed v orticit y analysis. Th us, when n udging to w

ard large-scale v orticit y elds, small-scale features in the model v orticit y will be selectiv ely damped. After all factors are considered, it is generally advisable to n udge the u- and v-componen ts of the winds directly whenev er possible (see Stauer and Seaman, 1990). Although this analysis-n udging tec hnique has been traditionally used to assimilate 3- D analyses based on ra winsonde observ ations, it can also be used to assimilate 2-D surface analyses within the model PBL (Stauer et al., 1991). While the t wice daily ra winsonde observ ations are spaced at about 400 km and n um ber appro ximately 100 o v

er the U.S., the spatial distribution of the surface data, a v

ailable at 3-h in terv als from the NCAR data arc hiv es, is considerably more dense. Depending on the time of da y

, there are roughly 500- 1200 surface data sites with an a v

erage spacing of about 100 km. The greater horizon tal and temporal resolution of con v

en tional surface data is especially attractiv e for mesoscale data assimilation. The surface-lev el analyses are assimilated in the same manner as the 3-D analyses, except for the v ertical exten t of their in
uence. These surface analyses, whic h are assumed to be represen tativ e at 10 m abo v

e ground lev el (A GL), should represen t only those scales resolv ed b y the model grid on to 28whic h they are assimilated. This is necessary to a v

oid diculties related to the insertion of small-scale div ergence patterns (near 2 x

) whic h migh t in teract adv ersely with the model's parameterizations (e.g., the moisture-con v

ergence criterion used to determine the existence and in tensit y of An thes-Kuo con v

ection). Eectiv e assimilation of single-lev el data depends on the equiv alen t depth o v

er whic h the data are inserted in to the model. Benecial eects on n umerical forecasts can be ac hiev ed b y distributing single-lev el data throughout sev eral model la y

ers. This approac h requires that the data be applied in a consisten t manner suc h that they are assured to be represen tativ e of those la y

ers. For example, the homogenizing eect of v ertical mixing during free con v

ectiv e conditions allo ws us to assume that surface-la y

er wind and mixing ratio (q) observ ations can be applied throughout the model PBL according to a conceptual model of boundary-la y

er structure. Suc h an idealized conceptual model is giv en b y Garratt et al. (1982) and is based on t ypical moderate to large instabilities observ ed at Wangara and Minnesota (Fig. 4.1). Ho w

ev er, in this same situation the frequen t presence of a shallo w superadiabatic la y

er near the surface mak es surface temperature or poten tial temperature data poorly represen tativ e of the boundary la y

er as a whole, and similarit y relationships describing the poten tial temperature prole become inaccurate. The same is true during nocturnal in v

ersion conditions. These and other factors mak e assimilation of single-lev el surface temperature observ ations unattractiv e for dening the temperature of the mixed la y

er abo v

e the surface la y

er. For example, n udging to w

ards an inaccurately diagnosed mixed-la y

er temperature can cause serious errors in the sim ulated PBL depth or ev en lead to a sudden spurious collapse of the unstable PBL structure. This can result from assimilating a surface temperature observ ation whic h is cooler than the model-sim ulated v alue b y only a few ten ths of one degree. In general, surface temperature data should not be directly assimilated in to the model (see Stauer et al., 1991). Figure 4.1 sho ws the unstable lo w

er troposphere comprised of three distinct la y

ers: a surface la y

er extending to heigh t

h

s

,aw ell-mixed la y

er from h

s

to h

1

and a transition la y

er extending from h

1

to h

2

. With the x-axis dened parallel to the mean wind, Fig. 4.1 suggests that the v componen

t of the wind is zero and there is th us no directional shear through the lo w

er t w

o la y

ers. The surface wind speed analysis, discussed abo v

e and 29

# h2 

# h1



# v u θHEIGHT



# MEAN WIND SPEED MEAN POTENTIAL TEMPERATURE 0 hs


# Figure 4.1 Schematic representation of mean wind speed and potential temperature profiles in an idealized conceptual model of the unstable atmospheric boundary layer.

-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.0 1.1 1.2 1.3 1.4 1.5 L = -100 m L = -10 m average fit

u40/u10=1.000+0.320*z0**0.200: u40/u10=1.169+0.315*z0: 0<z0<0.2 z0>0.2

_
z0 (m)u40/u10



# Figure 4.2 Schematic representing the relationship used to adapt the 10-m wind to the 40-m level as a function of roughness length (z0

# ) and Monin length (L).
30assumed to apply at 10 m A GL, can be modied via similarit y theory to apply at h s

. This modied wind for h

s

is represen tativ e of the mean mixed la y

er up to h

1

. These assumptions allo w the surface-la y

er wind information to be applied throughout the model's m ultila y

er PBL rather than at a single lev el. Similarly , the surface-la y

er mixing ratio (not sho wn) is also assumed to be represen tativ e of the en tire mixed la y

er, and ma y be applied o v

er sev eral model la y

ers during free-con v

ectiv e conditions. The surface-la y

er wind analyses, ho w

ev er, m ust be adapted for the depth of the surface la y

er, h

s

, whic h is assumed to be the heigh t of the lo w

est model la y

er under unstable conditions (40 m A GL for  = 0 : 995). Similarit y relationships that assume a logarithmic prole of wind with heigh t are used to adjust the 10-m surface wind analysis to the lo w

est model-la y

er heigh t. Figure 4.2 sho ws that this wind adjustmen t is more strongly dependen t on roughness ( z

0

) than on stabilit y as measured b y the Monin length ( L

); therefore, a \best- t" relationship is determined for a v

erage stabilit y conditions for roughness lengths ranging from 0.0 to 1.0 m. Th us, during free-con v

ectiv e conditions, the mean wind assimilated throughout the model PBL is based on the surface wind analysis modied to accoun t for the model surface-la y

er heigh t and the grid-bo x roughness. The surface analysis of mixing ratio is assumed to be represen tativ e for the lo w

est model la y

er and throughout the mixed la y

er. During unstable conditions, the Blac k

adar sc heme ma y dev elop a PBL whic h extends through sev eral model la y

ers. The \n udging correction" to the wind eld at the lo w

est model la y

er is applied throughout the model PBL to sim ulate the conceptual model. Because this idealization is also closely reproduced b y the PBL sc heme (without n udging), the n udging strategy within the PBL is compatible with the boundary-la y

er ph ysics. The observ ed surface mixing ratio applied throughout the model mixed la y

er is adjusted if necessary to remo v

e an y supersaturation with respect to the curren t model-sim ulated temperature and moisture conditions for a giv en lev el and grid poin t. This prev en ts the moisture assimilation term from becoming an articial source of precipitation. During stable conditions, on the other hand, the heigh t of the PBL is dened to be that of the lo w

est model la y

er since the boundary la y

er is largely decoupled from the free troposphere abo v

e. 31Therefore, the 3-hourly surface-analysis n udging is also giv en b y (4.1.2), but the v ertical exten t of the n udging is con trolled b y the model- sim ulated PBL depth, with ^ 0

for wind and moisture adjusted as previously discussed abo v

e. The analysis condence factor, 

, for the 3-h surface analyses, is functionally dependen t on the spatial distribution of the surface observ ations used to produce the analysis. Ov er land it v aries from unit y at grid bo xes within one-half the prescribed radius of in
uence of a surface observ ation to 0.2 for grid bo xes outside the prescribed radius. The v ertical w eigh ting factor, w



, is dened as

w 

= w

R  + w

S   1

: (4 : 1

: 8) where w

R  and w

S  represen t

w

 for assimilation of 3-D ra winsonde and 2-D surface data, respectiv ely , and w

S  depends on the model-sim ulated PBL depth. The surface data are assimilated with full strength ( w

S  = 1 : 0) within the la y

ers dening the PBL and with reduced strength ( w

S  = 0 : 9) one la y

er abo v

e (in the transition la y

er). The v ertical w eigh ting function used to assimilate 3-D ra winsonde data is dened suc h that w

R  = 0 : 0 in the PBL, 0.1 in the transition la y

er and 1.0 aloft. During stable conditions, therefore, the surface data are applied with full strength only in the lo w

est model la y

er and with reduced strength one la y

er abo v

e. Both t ypes of analysis n udging generally assimilate temporally in terpolated gridded analyses; that is, ^ 0

in (4.1.2) is in terpolated in time, for example, from either 12-h 3-D analyses or 3-h 2-D surface analyses. Therefore, w

t

is usually set to unit y

, except when decreasing the n udging at the end of a dynamic-initialization period. 4.2 Observ ational Nudging This alternativ e sc heme does not require gridded analyses of observ ations throughout the case study period, and ma y be better suited for situations with high-frequency asynoptic data (e.g., prolers), especially on the subalpha scales. Its form is similar to (4.1.2) and it uses only those observ ations whic h fall within a predetermined time windo w that is cen tered about eac h model time step. The set of dierences bet w

een the model and the observ ed state is computed at the observ ation locations, and analyzed bac k to the grid in a region surrounding the observ ations. The tendency for (

x

; t

) with G

p



= 0 is giv en 32b y @p  @ t = F

(

; x

; t

) + G 

p



P N i =1 W 2 i

(

x

; t

)

 i



( o ^

)

i P N i =1 W i (

x

; t

) ; (4 : 2

: 1) where F and G are as dened earlier, subscript i

denotes the i

th observ ation of a total of N whic h are within a preset radius of a giv en grid poin t, o

is the locally observ ed v alue of , and ^ is the model's prognostic v ariable in terpolated to the observ ation location in three dimensions. The observ ational qualit y factor, , whic h ranges from 0 to 1, accoun ts for c haracteristic errors in measurem en t systems and represen tativ eness. The four-dimensional w eigh ting function accoun ts for the spatial and temporal separation of the i

th observ ation from a giv en grid poin t at a giv en time step. The four-dimensional w eigh ting function for eac h observ ation i

in (4.2.1) is rewritten as W (

x

; t

) = w

xy 

w





w

t (4 : 2

: 2) The horizon tal w eigh ting function, w

xy , is a Cressman-t ype spatial w eigh ting function dened b y w

xy = R

2 D 2 R 2

+ D 2 0

 D  R (4 : 2

: 2) and w

xy = 0 D > R; (4 : 2

: 3) where R is the radius of in
uence and D is the distance from the i

th observ ation location to the grid poin t. The v ertical w eigh ting function, w



is also a distance-w eigh ted function dened b y w

 = 1 j 

obs 

j R  j 

obs 

j  R

 (4 : 2

: 4

a

)
w 

= 0 j 

obs 

j

> R 

; (4 : 2

: 4

b

) where R

 is the v ertical radius of in
uence and 

obs is the v ertical position of the i

th observ ation. The temporal w eigh ting function is giv en b y
w t

= 1 j t t

0

j

< = 2 (4 : 2

: 5)

w t

=  j t t

0

j  = 2  = 2

 j t t

0

j   (4 : 2

: 6) 33w t

= 0 j t t

0

j

> ; (4 : 2

: 7) where t

is the model-relativ e time, t

0 is the model-relativ e time of the i

th observ ation, and  is the half-period of a predetermined time windo w o v

er whic h an observ ation will in
uence the model sim ulation. For econom y

, the m ulti-lev el observ ations (soundings) used for obs n udging are usually v ertically in terpolated to the model sigma surfaces at eac h observ ation location prior to eac h sim ulation. Although the v ertical componen t of the w eigh ting function, w

 (4.2.4), is also a distance-w eigh ted function, the v ertical radius of in
uence, R

 , can be dened to be small (less than the spacing of the model la y

ers) so that eac h observ ation abo v

e the model surface la y

er in
uences only one sigma la y

er at a giv en location. Figure 4.3 illustrates sc hematically the horizon tal and temporal componen ts of W used for n udging to observ ations. The horizon tal w eigh ting function, w

xy , is the Cressman function giv en b y (4.2.2) and (4.2.3). As sho wn in the top of the gure, the horizon tal radius of in
uence v aries linearly in the v ertical with pressure, from R

s

at the surface to the preset v alue R

0

at a pressure lev el p

0

represen ting the free atmosphere, where terrain in
uences are assumed to be small. A

t pressures less than or equal to this user-dened v alue, dened b y default as 500 m b, the horizon tal radius of in
uence is dened b y default as t wice the v alue used in the surface la y

er, R

s

. For example, if R

s

= 100 km, R

0

= 200 km. For certain situations, suc h as with up w

ard propagating moun tain-induced gra vit y w a

v

es, the assumption of negligible terrain in
uence within the troposphere is in v

alid and should be a v

oided. As sho wn in the top of Fig. 4.3, the corrections computed at a giv en observ ation site and v ertical lev el abo v

e the surface la y

er (lo w

est model la y

er) are spread laterally along a constan t pressure lev el and th us across sev eral sigma la y

ers in regions of sloping terrain. That is, for an y giv en grid poin t within the horizon tal radius of in
uence, the obs-n udging correction in the horizon tal direction is applied to the sigma la y

er whic h has a pressure v alue closest to that of the observ ation. Observ ations within the model surface la y

er are spread along constan t sigma surfaces, but with a modied Cressman function (dashed con tours in the middle of Fig. 4.3) whic h reduces the in
uence of an observ ation as a function of the surface pressure (terrain). 34

# * * * * * * * * ****** * * * *

* * * ** * R

Rs 

# A σ1 

# σ2 

# σ3 

# σ4 

# σ5



# Z

X 

# P4

# ( P3 

# P2 

# P1 

# B P ) 

# * ( ( () ) )

Wxy


# Y

X

# A

z z+∆z z+∆zz+∆z z-∆z z z z B

Rs 

# 1.0 Wt



# 0.0 t0

# -

# τ t0 

# t0

# +

# τ Figure 4.3: Schematic showing the horizontal weighting function,w xy

# , and the temporal weighting function,wt

# , used for obs nudging.
35Th us spreading the in
uence of surface-la y

er observ ations along the lo w

est sigma ensures that the FDD A forcing near the surface in unev en terrain is con tin uous, and not lik e a pebble skipping across a pond. For observ ations in the surface la y

er, the distance factor D in (4.2.2) is replaced with D

m

, D m = D + R

S

C 1 m

j p

s

0 p

s

j ; (4 : 2

: 8) where D is as dened abo v

e, C

m is a constan t, and p

s

0

and p

s

are the surface pressures at the observ ation location and the grid poin t, respectiv ely . For example, C

m is t ypically dened as 75 m b, and R

s

is the surface-la y

er v alue for the horizon tal radius of in
uence. As the dierence in surface pressure bet w

een the observ ation location and the grid poin t approac hes C

m

, the second term in (4.2.8) approac hes R

, and w

xy tends to zero faster for a giv en D . Therefore, the eect of assimilating surface-la y

er observ ations in the v alley (moun tains) on grid poin t locations in the moun tains (v alley) will be m uc h reduced. This minimizes the possibilit y that observ ations in complex terrain will in
uence the model solution in areas where they ma y not be represen tativ e. Also, the v ertical w eigh ting factor, w 

, for these surface-la y

er observ ations is dened so that the v ertical in
uence of the surface-la y

er observ ations decreases linearly through the lo w

est 3 or so model la y

ers (about 250 m A GL). As men tioned earlier, single-lev el data are retained better b y the model if assimilated through sev eral v ertical la y

ers. The temporal w eigh ting function, w

t

(4.2.5-4.2.7), sho wn in the bottom of Fig. 4.3, is nonzero during a preset time windo w cen tered about the observ ation time, t

0

. It determines the time period o v

er whic h the ith observ ation can in
uence the model sim ulation via (4.2.1). In general, this time windo w can also be dened as a function of the pressure lev el of the observ ation similar to the eect of the horizon tal radius of in
uence, R

, in (4.2.2). Th us the nal correction to the model solution via obs n udging re
ects a w eigh ted a v

erage of all observ ations during the preset time windo w about the curren t time step and within some three-dimensional neigh borhood of eac h grid poin t.

365. Treatmen t of ph ysical processes 5.1 Horizon tal diusion Tw o t ypes of diusions are used to con trol nonlinear instabilit y and aliasing. These are a second-order diusion of the form F H 2 = p



K H

r 2 

; (5 : 1

: 1) where is an y prognostic v ariable, and a more scale-selectiv e fourth-order form F H 4 = p



K 0 H

r

4 

; (5 : 1

: 2) where K 0 H =  s

2

K H (5 : 1

: 3) ; The second order diusion is only used in the coarsest domain for the ro w and column of the grid poin ts next to the lateral boundaries, while the fourth-order form is used in the in terior of the coarsest domain as w ell as in the en tire domain of an y renemen t mesh. The horizon tal diusion coecien t K H consists of a bac kground v alue K H

0 and a term proportional to the deformation D K H = K H

0

+ : 5

k

2

 s

2

D (5 : 1

: 4) where k is the v on Karman constan t and D is giv en b y (Smagorinski et al. 1965) D = "



@u @x @v @y 

2

+ 

@v @x + @u @y 

2

#

1 2

: (5 : 1

: 5) A bac kground v alue of K H is a function of grid size and time step, where K H

0

= 3 :

10 3

 x

2  t : (5 : 1

: 6) Note that the horizon tal operators r

4

and r 2

are applied on constan t sigma surfaces. To ensure computational stabilit y

, an upper limit of 

x

2 64 t

is imposed on K H 5.2 Dry Con v

ectiv e Adjustm en t There ma y be situations in whic h super-adiabatic la y

ers are produced in the model atmosphere. When this happens, and there is no call to the Blac k

adar planetary boundary 37la y

er parameterization, a simple sc heme is used to remo v

e an y unstable la y

ers. The sc heme operates on the en tire sounding at once and conserv es the v ertical in tegral of in ternal and poten tial energy . When the model lapse rate of poten tial temperature @ @p exceeds a critical v alue 

@ @p 

c

, the sounding is adjusted so that (1) mass-w eigh ted mean temperature is unc hanged, and (2) the poten tial temperature lapse rate after adjustmen t equals 

@ @p 

c

. Giv en n la

y

ers in whic h the model poten tial temperature lapse rate exceeds the critical v alue, the rst constrain t giv es ( T n +  T n ) 

n

+ ( T n 1

+  T n 1

) 

n 1

+ ::: + ( T 1 +  T 1 ) 

1

=  T n X i

1

 

i

; (5 : 2

: 1) where T i are the adjustmen ts to be added to the temperature at la y

er i

,

T i and 

i

are the temperature and thic kness of the sigma la y

ers, and  T is the mass w eigh ted mean temperature. The second constrain t giv es ( T i +  T i )



i (

T i 1

+  T i 1

)



i 1

= 

@ @p 

( p

i p

i 1

) i

= 2 ; :::; n; (5 : 2

: 2) where 

i

is the Exner function. There are n equations that can be solv ed for n v

ariables  T i . The Gaussian elimination method is used to solv e the nn matrix system. After adjustmen t, the en tire sounding is rec hec k

ed for unstable la y

ers. The moisture in the adjusted la y

ers is assumed constan t in the v ertical, i.e., q vi =  q v

= P n i =1 q

vi

 

i P n i =1  

i (5 : 1

: 3) 5.3 Precipitation ph ysics MM5 has man y dieren t c hoices to treat precipitation ph ysics. They are usually divided in to t w

o dieren t groups: explicit and implicit sc hemes. Explicit sc hemes treat resolv ed precipitation ph ysics while implicit sc hemes treat the non-resolv ed precipitation ph ysics. Both ma y be operating at a grid-poin t at the same time. A common ly used terminology of \con v

ectiv e" v ersus \stable" precipitation is generally not acceptable on ner grid-resolutions, where con

v

ectiv e precipitation is quite often resolv ed. Hence in the follo wing subsections w e will use resolv ed/non-resolv ed and explicit/implicit as common terminologies. As t w

o additional options, MM5 allo ws for dry runs, where moisture is 38treated as a passiv e v ariable (no explicit and implicit sc hemes are applied). Another option is a \fak e dry run", where only the eects of the laten t heat release are remo v

ed. These 2 options do not require an y further description and will not be discussed in the follo wing subsections. 5.3.1 Resolv able scale precipitation processes These sc hemes are usually activ ated whenev er grid-scale saturation is reac hed. In other w ords, they treat resolv ed precipitation processes. The most simple w a

y that sometimes is still used on larger-scales, is to simply remo v

e super-saturation as precipitation and add the laten t heat to the thermodynam ic equation. More sophisticated sc hemes carry additional v ariables suc h as cloud and rain w

ater (subsection 5.3.1.1), or ev en ice and sno w (subsection 5.3.1.2). Both sc hemes described next are enhancemen ts of MM4's original sc heme (Hsie 1984). 5.3.1.1 Explicit treatmen t of cloudw ater, rain w

ater, sno w, and ice This sc heme optionally allo ws for ice-phase processes belo w 0 

C, where cloud w ater is treated as cloud ice and rain is treated as sno w (Dudhia 1989). The equations for w ater v apor, cloud w ater (ice) and rain w ater (sno w) mixing ratios are giv en b y the follo wing @p 

q

v @ t = m 2



@p 

uq v

=m @x + @p 

vq v

=m @y  @p 

q

v

_  @ + 

nh q

v

DIV + p



( P RE PCON P I I P ID ) + D

qv ; (5 : 3

: 1

: 1

: 1) @p 

q

c @ t = m 2



@p 

uq c

=m @x + @p 

vq c

=m @y  @p 

q

c

_  @ + 

nh q

c

DIV + p



(

P ID + P I I P RC P RA + PCON ) + D

qc ; (5 : 3

: 1

: 1

: 2) @p 

q

r @ t = m 2



@p 

uq r

=m @x + @p 

vq r

=m @y  @p 

q

r

_  @ + 

nh q

r

DIV @V f gq r @ + p



(

P RE + P RC + P RA ) + D

qr ; (5 : 3

: 1

: 1

: 3) where PCON is condensation (and freezing for T < 0



C) of w ater v apor in to cloud (ice) at w ater saturation, P RA is accretion of cloud b y rain (ice b y sno w), P RC is con v

ersion of cloud to rain (ice to sno w) and P RE is ev aporation (sublimation) of rain (sno w). Additional ice processes are P I I , the initiation of ice crystals, and P ID sublimation/deposition of cloud 39ice (Fig. 5.1). The fall speed of rain or sno w is V f . The term 

nh is 1 for nonh ydrostatic and 0 for h ydrostatic sim ulations. In all the relev an t processes, Marshall-P almer size distributions are assumed for rain and sno w and droplet fall speeds are tak en to be of the form V

(

D ) = aD b

, where D is the diameter. F or rain, the Marshall-P almer in tercept parameter is N

0 = 810 6

m 4

, a = 841 : 99667 and b

= 0 : 8 for V in m s 1

and D in meters, while for sno w N

0

= 210 7 m 4

,

a = 11 : 72 and b

= 0 : 41. The saturated v apor pressure o v

er w ater (in m b) is tak en to be
e sw = 6 : 112exp 

17 : 67 

T 273 : 15 T 29 : 65  ; (5 : 3

: 1

: 1

: 4) and for ice e

si = 6 : 11exp 

22 : 514 6150 T 

: (5 : 3

: 1

: 1

: 5) The saturated w ater v apor mixing ratio is then giv en b y q s = 0

: 622 e

s p e

s

: P RC , the autocon v

ersion term is represen ted b y P RC = max [ k

1

(

q

c q

crit )

;

0]

; (5 : 3

: 1

: 1

: 6

a

) for cloud to rain and b y P RC = max [(

q

c Mmax n

c

)

=



t; 0]

; (5 : 3

: 1

: 1

: 6

b

) for ice to sno w, where k

1

= 10 3

s 1

,

q

crit = 0 : 5 g kg 1

,

Mmax = 9 : 410 10 kg and n

c is giv en b y Fletc her's (1962) form ula for the n um ber concen tration of ice n uclei (kg 1

),
n c = 10 2

exp[0 : 6(273 : 15 T

)]

=: P I I , the initiation of ice crystals is giv en b y P I I = max [(

M0 n

c q

c

)

=



t; 0]

; (5 : 3

: 1

: 1

: 7) as long as sucien t supersaturation o v

er ice exists, where M0 = 10 12 kg. 40

# ice cloud

vapor

snow rain PRF

PRA PRC PRA PRC PID PII PCON

PRM PMF PRE PRE

Figure 5.1 Box diagram illustrating the processes in the moisture scheme for ice (crystals), cloud(liquid), snow and rain. PCON, condensation/evaporation of cloud; PRA, accretion; PRC, conversion; PID, deposition onto ice crystals; PRE, evaporation for rain and deposition/sublimation for snow; PMF, melting/freezing due to advection; PII, initiation of ice crystals; and PRM, melting of snow due to fall.

41P RA , the accretion rate is giv en b y P RA = 1 4

aq c

EN 0

(3 + b

)  3+ b ; (5 : 3

: 1

: 1

: 8) where  is the gamm a-function, E is the collection eciency (1 for rain and 0.1 for sno w) and  is giv en b y  = 

N0 

w q r 

1

=

4

: Here 

w is the mean densit y of rain or sno w particles (1000 and 100 kg m 3

, respectiv ely .) P ID , the deposition on to or sublimation of ice particles is found from P ID = 4

D

i

(

S

i 1) n c A + B ; (5 : 3

: 1

: 1

: 9) where S

i

= q

v

=q si

; A = L

2 s

 K a

R

v

T

2 ; B = 1 q si



: L s

is the laten t heat of sublimation, K a

is the thermal conductivit y of air, R

v

is the gas constan t for w ater v apor, and  is the diusivit y of v apor in air. The mean diameter of ice crystals, D

i

, is found from the mean mass, Mi = q

c

=n c

, and the mass-diam eter relation for hexagonal plates from Rutledge and Hobbs (1983), D

i

= 16 : 3

M 1

=

2 i

meters. P RE , the ev aporation of rain and sublimation/deposition of sno w can be determined from P RE = 2

N0 (

S 1) A + B "

f1 

2 + f2 

a  

1

=

2

S

1

=

3 c

(5 =

2 + b= 2)  5

=

2+ b=

2 #

; (5 : 3

: 1

: 1

: 10) with the relev an t

N0 ,

a

, and b

c

hosen for rain or sno w, and S = S

w or S

i

. The denition of A and B also c hange from the abo v

e for rain, substituting L

v

for L

s

and q

sw for q

si

. For sno w, 2  is replaced b y 4. The v alues of f1 and f2 are 0.78 and 0.32 for rain and 0.65 and 0.44 for sno w. The term in brac k

ets represen ts a distribution-in tegrated v en tilation factor, F = f

1

+ f2 S

1

=

3 c

Re 1

=

2

, with S

c

= = , the Sc hmidt n um ber, and Re = V

(

D )

D= , the Reynolds n um ber, and  is the dynamic viscosit y of air. 42PCON , the condensation is determined as follo ws. Temperature, w ater v apor mixing ratio and cloud w ater are forecast rst: these preliminary forecast v alues are designated b y T



,

q

 v

and q

 c

. We dene M = q

 v q

 vs ; where q

 vs is the saturated mixing ratio at temperature T



, (1) if M > 0 (supersaturation), PCON = r

1

M 

t ; (5 : 3

: 1

: 1

: 11 a

) where r

1 = 1 1 +

L

2 v

q

 vs R v

c

pm T



2

; (2) if M < 0 and q

c > 0 (ev aporation), PCON = min  r

1

M 

t ;

q

 c  t



; (5 : 3

: 1

: 1

: 11 b

) (3) if M < 0 and q

c = 0, PCON = 0 : (5 : 3

: 1

: 1

: 11 c

) The PCON term is computed diagnostically so no iteration is needed. Additionally , as sno w falls through the 0 

C lev el, it immediately melts to rain. This process is giv en b y P RM = gV f q

r  p : (5 : 3

: 1

: 1

: 12) Adv ection of ice or sno w do wn w

ards or of rain or cloud up w

ards through this lev el also melts or freezes the particles, where P MF = !

(

q

c + q

r

)  p : (5 : 3

: 1

: 1

: 13) In both cases, the 0 

C isotherm is tak en to be at a full model lev el boundary . Melting occurs at the lev el immediately belo w this boundary and freezing abo v

e it. The laten t heating is th us _ Q = L

(

P RE + P ID + P I I + PCON ) + L

m

(

P RM + P MF )

; (5 : 3

: 1

: 1

: 14) 43where L = L

v

for T > 0 

C and L = L

s

for T < 0 

C, and L

m = L

s L

v

. The fall speed is mass-w eigh ted and so is determined from V f = a

(4 + b

) 6

 b

: (5 : 3

: 1

: 1

: 15) The fall term in (5.3.1.1.3), the rain and sno w prediction equation, ma y be calculated on split time-steps,  t

0 , in the explicit moisture routine. This ensures that V f 

t

0 =

 z < 1, whic h is required for n umerical stabilit y

. The size of  t

0

is determined independen tly in eac h model column based on the maxim um v alue of V f 

t=  z

in the column, where  t

is the model time step.

445.3.1.2 Mixed-Phase Ice Sc heme This sc heme is based on the simple ice phase sc heme described in the previous subsection, but it does not immediately freeze or melt w ater and ice. Supercooled w ater can exist belo w 0 

C in this sc heme, as can unmelted sno w exist abo v

e 0 

C. Separate arra ys are used to store v apor, cloud, rain, cloud ice and sno w. Homogeneous freezing of cloud w ater to cloud ice occurs immediately belo w -40 

C and cloud ice melts immediately abo v

e 0 

C. Sno w melts according to PSM = 2

N 0

s L f K a

(

T T 0 )

"

f1 

2 + f2 

a  

1

=

2

S

1

=

3 c

(5 =

2 + b= 2)  5

=

2+ b=

2 #

; (5 : 3

: 1

: 2

: 1) where f

1

= 0 : 78 and f2 = 0 : 31 (Rutledge and Hobbs 1984), and the other constan ts are the ones relev an t to sno w in subsection (a). Ev aporation of melting sno w is modied to use the v alues of A and B for rain as in (5.3.1.1.10). Heterogeneous freezing of cloud w ater to cloud ice is also included follo wing Bigg (1953), PCI = B

0 f

exp[ A

0 (

T 0 T

)] 1

g q 2 c  w

Nc (5 : 3

: 1

: 2

: 2) where A

0

= 0 : 66K 1

,

B

0

= 100m 3

s 1 and the n um ber concen tration of cloud droplets per unit v olume of air, Nc = 10 10

m 3

. Sekhon and Sriv asta v

a (1970) determined that better comparison against observ ed sno w distributions can be obtained in theoretical studies if the slope in tercept v alue for the size distribution is expressed as N0 s

(

m 4

)=1 : 05 R 0

: 94 (5 : 3

: 1

: 2

: 3) where, N

0

s

is the slope in tercept and R (m s 1

) is the sno w fall rate. Th us a v ariable in tercept parameter replaces the constan t

N

0

s

used in the simple ice sc heme. This can be expressed in terms of sno w mixing ratio, q

S

, as N0 s

= 8 < :

1

: 05 " 1 q S 

 S q S

 b 4

#

0

: 94 9 = ; 4 0 : 94

b

+4 (5 : 3

: 1

: 2

: 4) where, 1

= 6



w a (4+ b

)

. 455.3.2 Im plicit cum ulus param eterization sc hemes 5.3.2.1 The Kuo sc heme In this sc heme, the amoun t of con v

ection is determined b y the v ertically in tegrated moisture con v

ergence. The feedbac k to the larger scale (the v ertical distribution of heating and moistening), is determined with the help of the normalized v ertical proles of con v

ectiv e heating ( N

h

(



)) and moistening ( Nm (



)), and a v ertical eddy-
ux div ergence of w ater v apor associated with cum ulus con v

ection V qf (



). Therefore, equations (2.1.3), (2.2.5) and (5.3.1.1.1) can be rewritten to include the con v

ectiv e-scale 
uxes as @p 

T @ t = m 2



@p 

uT =m @x + @p 

vT =m @y  @p 

T _  @ + p

 ! c

p + p



L

v c pm N

h

(



)(1 b

) gM t + D

T

; (5 : 3

: 2

: 1

: 1) @p 

T @ t = m 2



@p 

uT =m @x + @p 

vT =m @y  @p 

T _  @ + T :DIV + 1 c

p  p



Dp 0 Dt 

0

gp 

w D

p

0  + p



L

v c pm N

h

(



)(1 b

) gM t + D

T

;

(5 : 3

: 2

: 1

: 2) @p 

q

v @ t = m 2



@p 

uq v

=m @x + @p 

vq v

=m @y  + 

nh q

v

DIV + p



( P RE PCON P I I P ID ) + p



bgM t Nm (



)+ p



V qf (



) + D

qv ;

(5 : 3

: 2

: 1

: 3) where the v ertically in tegrated moisture con v

ergence Mt is M t = 

m 2 g

 Z 1 0

r _ p 

V q v m

d: (5 : 3

: 2

: 1

: 4) A portion (1 b

) of M t is assumed to condense and precipitate, where the remaining fraction b

is assumed to moisten the grid column. Follo wing An thes(1977), b

is a function of the mean relativ e h umidit y RH of the column, where b = 2(1 RH ) (5 : 3

: 2

: 1

: 5) for RH  0

: 5, and b

= 1 otherwise. The v ertical proles of heating and moistening 46The normalized, nondimensional functions for the v ertical proles of heating and moistening and the div ergence of the v ertical eddy 
ux of w ater v apor are sub ject to the constrain ts Z 1 0

N

h

(



)

d = 1 ; (5 : 3

: 2

: 1

: 6) Z 1 0

Nm (



)

d = 1 ; (5 : 3

: 2

: 1

: 7) Z 1 0

V qf (



)

d = 0 : (5 : 3

: 2

: 1

: 8) An thes et al. (1987) assume simple relationships for these functions, whic h are deriv ed from budget studies. F

or the con v

ectiv e heating prole, N

h

, they observ e that the con v

ectiv e heating often has a parabolic shape with a maxim um in the upper half of the cloud. Hence N

h

(



) = a

1

x

2

+ a

2

x

+ a

3

; (5 : 3

: 2

: 1

: 9) where x = ln (5 : 3

: 2

: 1

: 10) with the boundary conditions: N h

(



)=0 ; at x

b

= ln b

; and x u = ln u (5 : 3

: 2

: 1

: 11) at cloudbase ( 

b

) and cloud top ( 

u

), and N 0 h

(



) = @N h

(



) @

= 0 (5 : 3

: 2

: 1

: 12) at  x , whic h is dened as  x

= x

u

+ x

b 2

; (5 : 3

: 2

: 1

: 13) where subscripts u and b

stand for the top and the base of the cloud. Using (5.3.2.1.6), a

1 can be sho wn to be a

1

= 2 x 3 u x

3 b

+ x

2 u

x

b x

u

x

2 b

: (5 : 3

: 2

: 1

: 14) The v ertical moistening prole, N

m

(



), is simply giv en follo wing An thes (1977) as Nm (



) = (1 RH (



))

q

v

s

(



) R 1  ktop (1 RH (



0 ))

q

vs (



0 )

d 0

: (5 : 3

: 2

: 1

: 15) 47Div ergence of the Vertical Eddy Flux of Water Vapor V qf (



) The div ergence of the v ertical eddy 
ux of w ater v apor is dened as V qf (



) = @

_ 

0 q

0 v @

: (5 : 3

: 2

: 1

: 16) If one assumes a small fraction of con v

ectiv e cloud co v

er, and the cloud v ertical motion _  c is m uc h larger than the larger-scale v ertical motion, _  (5.3.2.1.16) can be rewritten as V qf (



) = a 1 a @ @ [ _ 

c

(

q

vc q

v

)]

; (5 : 3

: 2

: 1

: 17) where q

vc is the mixing ratio in the cloud. According to An thes (1977), the fractional co v

erage a

is calculate using

a = (1 b

) gM t R p

 0

 !

c

@q

vc @p + @q

vc @t e



dp ; (5 : 3

: 2

: 1

: 18) whic h is the ratio bet w

een the grid-a v

erage condensation rate and that of a single cloud. The term @q

vc @t e represen ts the con tribution to the rate of c hange of cloud-mixing ratio b y en trainmen t (An thes 1977). An thes et al. (1987) assume a t ypical v alue for the denominator of appro ximately 4 : 3

10 3

cb s 1

and then rewrite (5.3.2.1.17) as V qf (



) = (1 b

) gM t 4 : 3

10 3 @ @ [ _ 

c

(

q

vc q

v

)]

: (5 : 3

: 2

: 1

: 19) F or further simplication, An thes et al. (1987) next assume that _  c

also has a parabolic shape and can be expressed as _ 

c

= c

1

x

2

+ c

2

x

+ c

3

; (5 : 3

: 2

: 1

: 20) where x = lnp , and _  c

= 0 at cloud-top and base. Furthermore, q

vc q

v

is assumed to ha v

e a parabolic prole with pressure q vc q

v

= b

1

x

2

+ b

2

x

+ b

3 (5 : 3

: 2

: 1

: 21) with x = ln [(1 

)(100 p

t

) + p

t

]

: (5 : 3

: 2

: 1

: 22) 48The procedure The simple procedure can be summ arized as follo ws: 1. Compute M t from (5.3.2.1.4) 2. Chec k whether M t  3

:

10 5

kg m 2

s 1

, a critical threshold v alue. 3. Chec k the model sounding for con v

ectiv e instabilit y to see if con v

ection is possible. 4. Determine cloud top and base from sounding. 5. Chec k whether cloud-depth is larger than a critical v alue (   : 3) 6. Calculate the normalized v ertical prole functions 7. Calculate _  c

on the full  lev els from (5.3.2.1.20) 8. Compute q

vc q

v

from (5.3.2.1.21) 9. Calculate V qf from (5.3.2.1.19)

495.3.2.2 A modied Arak a

w

a-Sc h

ubert sc heme The v ersion of the Arak a

w

a-Sc h

ubert sc heme used here w as dev eloped b y Grell (1993). In con trast to the original sc heme (Arak a

w

a and Sc h

ubert 1974, AS), it includes moist con v

ectiv e-scale do wndrafts. Other c hanges ha v

e been implemen ted to also allo w the sc heme to be used successfully in mesoscale models in mid-latitudes (Grell et al. 1991). To simplify the description w e ha v

e adapted a terminology originally in troduced b y Betts (1974), whic h splits the parameterization problem from the modeling view in three parts: static con trol, dynamic con trol, and feedbac k. The static con trol includes usually a cloud-model and calculates cloud thermodynam ic properties, the dynamic con trol is what determines the amoun t and location of the con v

ection, and the feedbac k determines the v ertical distribution of the in tegrated heating and moistening. Static con trol As with all commonly used one dimensional steady state cloud models (plumes, bubbles, or jets), our AS sc heme mak es use of the assumption that en trainmen t occurs o v

er the depth of the buo y

an t elemen t according to the en trainmen t h ypothesis  = 1 m (

z

)

@m (

z

) @z

 : 2 r

; (5 : 3

: 2

: 2

: 1) where  is the total net fractional en trainmen t rate of the buo y

an t elemen t, m its mass 
ux ( m u

for updraft, m d

for do wndraft), and r

its radius. F ollo wing AS, the dependence on the radius is not explicitly used. Ho w

ev er, implicitly , the radius of the cloud is assumed to be constan t. Detrainmen t w as originally only assumed to happen at the cloud top, but this assumption ma y easily be v aried (Houze et al. 1979, Lord 1978) b y dening a fractional detrainmen t rate, 

ud , and rewriting (5.3.2.2.1) for the updraft of cloud t ype  as  u

= 

ue 

ud = 1 m u

(

z

)

@m u

(

z

) @z

= 1 m u

(

; z )

 @m u

(

; z ) @z



ent 

@m u

(

; z ) @z



det  (5 : 3

: 2

: 2

: 2)
where 

ue is the gross fractional en trainmen t rate, and 

u is the total net fractional en trainmen t rate of the updraft. Subscripts ent and det indicate c

hanges due to en trainmen t and detrainmen t, respectiv ely . Looking at the budget of a thermodynamic 50v ariable in an innitesimal la y

er of the updraft w e get @m u u @z = 

@m u @z



e

~ 

@m u @z



d u

+ S

u

: (5 : 3

: 2

: 2

: 3) Together with (5.3.2.2.2) this leads to the steady state plume equation @ u

(

; z ) @z

= 

ue ( ~ (

z

) u

(

; z )) + S

u (5 : 3

: 2

: 2

: 4) where is a thermodynam ic v ariable, the tilde denotes an en vironmen tal v alue, and subscript u denotes an updraft propert y

. S stands for sources or sinks. Similarly , for the do wndraft, w e can rewrite equations (5.3.2.2.2) and (5.3.2.2.4) as  d

= 

de 

dd = 1 m d

(

z

)

@m d

(

z

) @z

= 1 m d

(

z

)

 @m d

(

z

) @z



ent 

@m d

(

z

) @z



det  (5 : 3

: 2

: 2

: 5) and @ d

(

z

) @z

= 

de ( ~ (

z

) d

(

z

)) + S; (5 : 3

: 2

: 2

: 6) where subscript d denotes a do wndraft propert y

. For moist static energy ~ h (

z

) = Cp ~ T

(

z

) + gz + L

~ q

(

z

)

; (5 : 3

: 2

: 2

: 7) equations (5.3.2.2.4) and (5.3.2.2.6) simply become @h u

(

; z ) @z

= 

ue (

~ h

(

z

) h

u

(

; z )) (5 : 3

: 2

: 2

: 8) and @h d

(

z

) @z

= 

de [ ~ h

(

z

) h

d

(

z

)]

: (5 : 3

: 2

: 2

: 9) Next, for the moisture budget of the updraft, w e use

u = q

u

(

; z ) + q

l (

; z ) (5 : 3

: 2

: 2

: 10) and S

u

= c

0

m u

(

; z )

q

l

(

; z )

: (5 : 3

: 2

: 2

: 11) 51Here S

u is the total w ater that is rained out, c

0

is a rainfall con v

ersion parameter and could be a function of cloud size or wind shear, q

l

is the suspended liquid w ater con ten t of the cloud, and q

u

is the w ater v apor mixing ratio inside the updraft. Equation (5.3.2.2.4) can then be rewritten as @ (

q

u

(

; z ) + q

l (

; z )) @z = 

ue (~ q (

z

) q

u

(

; z ) q

l (

; z )) + S

u

: (5 : 3

: 2

: 2

: 12) For the do wndraft, the equation for the w ater v apor reads @q d

(

z

) @z

= 

de [~ q

(

z

) q

d

(

z

)] + S

d

: (5 : 3

: 2

: 2

: 13) S d

here is a source; namely the ev aporation of rain. Assuming saturation in the updraft and do wndraft, w e can mak e use of the appro ximate equation q c

(

; z )=~ q 

+ 1 + 1 L

[ h

c

(

; z ) ~ h



(

z

)]

; (5 : 3

: 2

: 2

: 14) where

= L c p



@

~ q

 @T



P (5 : 3

: 2

: 2

: 15) the asterisk denotes a saturated v alue, and h

c

here stands for the moist static energy in the cloud (updraft or do wndraft), if saturation is assumed. Next, to arriv e at a usable closure, the up- and do wn-draft mass 
uxes are normalized b y the updraft base ( m b

(



)) mass 
ux, and the do wndraft base m 0

(



) mass 
ux of a subensem ble. Hence, for the updraft,

m u

(

; z ) = m b

(



)



u

(

; z ) (5 : 3

: 2

: 2

: 16) and 

ue 

ud = 1  u

(

z

)

@ u

(

z

) @z

: (5 : 3

: 2

: 2

: 17) Equiv alen tly , for the do wndraft w e ma y write
m d

(

z

) = m 0

(



)



d

(

; z ) (5 : 3

: 2

: 2

: 18) and 

de 

dd = 1  d

(

z

)

@ d

(

z

) @z

: (5 : 3

: 2

: 2

: 19) 52Here, m 0

is the mass 
ux at the originating lev el and 

d

, as 

u

in equation (5.3.2.2.16), is the normalized mass 
ux prole. To lea v

e only one unkno wn v ariable, w e follo w Houze et al. (1979) and mak e the originating mass 
ux of the do wndraft a function of the updraft mass 
ux and reev aporation of con v

ectiv e condensate. Therefore, the condensate in the updraft C u

(



)

d = m b

d 

Z z

T
z B 

u

(

; z )

S

u

dz )

  I1 m b

d (5 : 3

: 2

: 2

: 20) is apportioned according to C u

(



)

d = ( R

c

(



) + E

d

(



))

d = ( (



) + (



))

C

u

(



)

d; (5 : 3

: 2

: 2

: 21) where + = 1 and E

d

, the ev aporation of condensate in the do wndraft for cloud t ype  , can be written as E d

d = m 0

(



)

d 

Z z

0 0



d

(

; z )

S

d

dz   I2 m 0

d: (5 : 3

: 2

: 2

: 22) F rom equations (5.3.2.2.20-5.3.2.2.22) w e see that E d

d = C u

d = I1 m b

d = I2 m 0

d (5 : 3

: 2

: 2

: 23) and hence m 0

(



) = (



)

I1 m b

(



) I2 (



) = 

(



)

m b

(



)

: (5 : 3

: 2

: 2

: 24) Here 1 is the precipitation eciency . Follo wing Fritsc h and Chappell (1980), it is made dependen t on the windshear. To solv e the abo v

e equations w e need to specify boundary conditions as w ell as mak e some arbitrary assumptions. For the updraft w e assume h u

(

z

b

) = MAX (

~ h

(

z

)) (5 : 3

: 2

: 2

: 25) with z

 z

b and h

u

(

; z T

) = ~ h



(

z

T

)

; (5 : 3

: 2

: 2

: 26) 53where the asterisk denotes a saturation v alue. Similarly , for the do wndraft, h d

(

z

0

) = MIN (

~ h

(

z

))

: (5 : 3

: 2

: 2

: 27) Ph ysically , for both updraft and do wndraft, w e allo w for maxim um buo y

ancy . The boundary conditions for the updraft are dieren t than in the original sc heme, whic h had a rigid dependence on the planetary boundary la y

er heigh t. In the original sc heme, the mixed la y

er w as assumed to be w ell mixed, and the cloud base w as located on top of the mixed la y

er. In semi-prognostic tests (Grell et al. 1991) large v ariations of moist static energy proles w ere found in v ery lo w lev els of the troposphere. This w as caused b y cold do wndraft out
o w. Naturally , the in
o w to an updraft will not be a mixture of do wndraft air and the more buo y

an t air; it is more lik ely the air with high moist static energy from the la y

er abo v

e the do wndraft out
o w. Furthermo re, compensatory subsidence should only con tin ue to the lev el from whic h the updraft dra ws its air. Compensatory uplifting ma y be required in v ery lo w la y

ers of the troposphere because of the do wndraft mass 
ux. F eedbac k The feedbac k to the larger-scale en vironmen t is expressed in a con v

enien t form as  @s @ t 

cu = 1

: 

@ @z F s Ll ; (5 : 3

: 2

: 2

: 28)  @q @ t 

cu = 1

: 

@ @z F q +

l R; (5 : 3

: 2

: 2

: 29) where s

is the dry static energy ( s

= c

p

T + gz ). The con v

ectiv e-scale 
uxes within a grid bo x are dened as F s Ll  F s LF l (5 : 3

: 2

: 2

: 30) F q +

l

 F q + F l (5 : 3

: 2

: 2

: 31) where F s is the 
ux of dry static energy ,

F q is the 
ux of w ater v apor, and F l is the 
ux of suspended cloud liquid w ater. These are dened as F s (

z

)

 + Z  

u

(

; z )[

s

u

(

; z )  s

(

z

)]

m b

(



)

d Z  

d

(

; z )[

s

d

(

; z )  s

(

z

)]

m 0

(



)

d (5 : 3

: 2

: 2

: 32)

54F q (

z

)

 + Z  

u

(

; z )[

q

u

(

; z )  q

(

z

)]

m b

(



)

d Z  

d

(

; z )[

q

d

(

; z )  q

(

z

)]

m 0

(



)

d (5 : 3

: 2

: 2

: 33)

F l (

z

)

 Z  

u

(

; z )

l

(

; z )

m b

(



)

d (5 : 3

: 2

: 2

: 34) The rainfall (con v

ectiv e-scale sink of cloud w ater) is dened as R (

z

)

 + Z  

u

(

; z )

c

0

(



)

l

(

; z )

m b

(



)

d Z  

d

(

; z )

q

e

(

; z )

m 0

(



)

d (5 : 3

: 2

: 2

: 35)

Here q

e

is the amoun t of moisture that is necessary to k eep the do wndraft saturated. The second term on the righ thand sides is due to do wndrafts and is zero abo v

e the downdraftoriginating lev el. Belo w the updraft-air originating lev el, the rst term on the righ t-hand sides is zero and only do wndrafts aect the larger-scale en vironmen t. Belo w the updraft-air originating lev el, the con v

ectiv e-scale 
uxes due to updrafts are zero. Bet w

een the updraftair-originating lev el and the lev el of free con v

ection (the LF C), F l and R are set to zero. Since no liquid w ater is assumed to be in the en vironmen t as the do wndraft, the do wn w

ard ux due to updrafts as w ell as do wndraft 
uxes in equation (5.3.2.2.33) are zero. Sc h

ubert (1974) sho w

ed that con v

ection will not increase the total moist static energy per unit area in a column. In essence, only precipitation can c hange the dry static energy budget and the total mass of w ater v apor. All v ariables in the 
ux terms can be determined from the equations for the static con trol, except m b

(



). This is determined in the dynamic con trol, whic h incorporates the closure assumption of the sc heme and is described next. Dynamic con trol Arak a

w

a-Sc h

ubert rst in troduced the cloud w ork function, whic h is an in tegral measure of the buo y

ancy force associated with a subensem b

le. Starting with dw u dt = B

u F r = dw u dz

dz dt = d dt d dz w

2 u 2 = 1 w

u d dt w

2 u 2

; (5 : 3

: 2

: 2

: 36) 55where B

u

is the acceleration due to buo y

ancy and F r the deceleration due to friction, and m ultiplying equation (5.3.2.2.36) b y 

u

(

; z )

w

u

(

; z ), giv es d dt 

u

w

2 u 2 = 

u

w

u

(

B

u F r )

: (5 : 3

: 2

: 2

: 37) In tegrating o v

er the depth of the updraft and using m u

= 

u

w

u

= m b



u

yields d dt Z z

T
z b 

u

w

2 u 2

dz = m b

(



)

Z z

T
z b 

u

B

u

dz D

u

; (5 : 3

: 2

: 2

: 38) where D is the updraft-scale kinetic energy dissipation. Equation (5.3.2.2.38) can be written in the sym bolic form d dt

KE u

= A

u

(



)

m b

(



) D

u

(



)

; (5 : 3

: 2

: 2

: 39) where A

u

(



) is a measure of the eciency of kinetic energy generation inside the cloud and is called the cloud w ork function. It can also be written as A u

(



) = Z z

T
z B g Cp T

(

z

)



u

(

; z ) 1 +

(

h

u

(

; z ) ~ h



(

z

))

dz ; (5 : 3

: 2

: 2

: 40) where is dened as in equation (5.3.2.2.15). As with equations (5.3.2.2.36-5.3.2.2.38), dening a kinetic energy generation inside the do

wndraft leads to d dt KE d

= A

d

(



)

m 0

(



) D

d

(



)

; (5 : 3

: 2

: 2

: 41) where A

d

, the measure of the eciency of kinetic energy generation inside the do wndraft, can be written as A d

(



) = Z z

sur z 0 g Cp T

(

z

)



d

(

; z ) 1 +

(

~ h



(

z

) h

d

(

; z ))

dz : (5 : 3

: 2

: 2

: 42) Note that dry static energy instead of moist static energy w ould ha v

e to be used if subsaturation is assumed. We can com bine equation (5.3.2.2.39) and (5.3.2.2.41) and then mak e use of (5.3.2.2.24) to yield d dt KE tot = A

tot (



)

m b

(



) D

tot (



)

; (5 : 3

: 2

: 2

: 43) 56where A

tot (



) = A

u

(



) + 

(



)

A

d

(



) (5 : 3

: 2

: 2

: 44) is the total cloud w ork function whic h w as redened as a measure of the eciency of kinetic energy generation in updrafts as w ell as do wndrafts. Next, AS separated the c hange of the cloud w ork function in to t w

o parts: One is due to the c hange in the larger-scale v ariables  dA tot dt 

LS  F

(



)

; (5 : 3

: 2

: 2

: 45) and one is due to the modication of the en vironmen t b y the clouds. Since the cum ulus feedbac k on the larger-scale elds is a linear function of m b

, this term can be written in the sym bolic form 

dA tot dt 

CU  Z  K (

;  0 )

m b

(



0 )

d: (5 : 3

: 2

: 2

: 46) Therefore dA tot dt = F

(



) + Z  K (

;  0 )

m b

(



0 )

d; (5 : 3

: 2

: 2

: 47) where K (

;  0 ) are the k ernels. The k ernels are an expression for the in teraction bet w

een clouds (updrafts and do wndrafts). Equation (5.3.2.2.47) is solv ed with a linear programming method (Lord 1978). In the original v ersion of the Arak a

w

a-Sc h

ubert sc heme, the fractional en trainmen t rate w as the parameter whic h c haracterized the cloud. In later papers, the cloud-top detrainmen t lev el w as c hosen instead. If a ne v ertical resolution is assumed, the second c hoice will most lik ely be better n umerically , since no in terpolation is necessary at the cloud tops. Ho w

ev er, in the extremely unstable en vironmen t of the mid-latitudes, it is sometimes impossible to calculate \clouds" with cloud tops in the unstable la y

ers. En trainmen t rates w ould ha v

e to be extremely large to stop cloud gro wth. We therefore c hose the fractional en trainmen t rate as the spectral parameter. The procedure The cloud base is a function of time and space. Ho w

ev er, at a specic grid poin t the cloud base will be the same for ev ery mem ber of the subensem ble. We also distinguish among an updraft-air originating lev el, z

u

, a do wndraft-air originating lev el, z

0

, a cloud 57base lev el, z

b

(the LCL), and a lev el of free con v

ection, z

bc (LF C). Here, z

u

is determined from condition (5.3.2.2.25) and determines the thermodynamic properties of the updraft from cloud t ype i

. The air becomes saturated at z

b

; condensation will start, but no con v

ection can occur y et because the buo y

ancy is negativ e. In some instances this lev el could be the same as the LF C. The LF C is of great importance since this is the lev el at whic h the static con trol starts the calculations of individual con v

ectiv e elemen ts. Since the air that feeds the cloud originates belo w the LCL, compensatory subsidence is allo w

ed to reac h the originating lev el of the updraft air. For the do wndraft, the originating lev el is also a function of time and space. If the do wndraft exists, it will alw a

ys reac h the surface. For updraft and do wndraft in la y

er k the mass budgets are dened as
e u

(

k; i ) d

u

(

k; i ) = 

u

(

k

+ : 5

; i

) 

u

(

k : 5

; i

) (5 : 3

: 2

: 2

: 48 a

) and e

d

(

k; i ) d

d

(

k; i ) = 

d

(

k

+ : 5

; i

) 

d

(

k : 5

; i

)

; (5 : 3

: 2

: 2

: 48 b

) where en trainmen t for the updraft and do wndraft is dened as
e u

(

k; i ) = 

ue  z

d



u

(

k

+ : 5

; i

) (5 : 3

: 2

: 2

: 49 a

)
e d

(

k; i ) = 

de  z

d



d

(

k : 5

; i

) (5 : 3

: 2

: 2

: 49 b

) and detrainmen t is dened as d

u

(

k; i ) = 

ud  z

d



u

(

k

+ : 5

; i

) (5 : 3

: 2

: 2

: 50 a

) d d

(

k; i ) = 

dd  z

d



d

(

k : 5

; i

)

: (5 : 3

: 2

: 2

: 50 b

) Com bining the abo v

e three equations for the updraft and do wndraft yields  u

(

k : 5

; i

) = 

u

(

k

+ : 5

; i

)(1 :

+ 

ue  z

d 

ud  z

d

) (5 : 3

: 2

: 2

: 51 a

) for the updraft and  d

(

k

+ : 5

; i

) = 

d

(

k : 5

; i

)(1 :

+ 

de  z

d 

dd  z

d

) (5 : 3

: 2

: 2

: 51 b

) 58for the do wndraft. Here w e dene  z

d

= z

(

k

+ : 5) z

(

k : 5). The discretized form for the do wndraft moist static energy budget reads
e d

(

k; i )

~ h

(

k

) d

d

(

k; i )

h

d

(

k

+ : 5

; i

) h

d

(

k : 5

; i

) 2
= 

d

(

k

+ : 5

; i

)

h

d

(

k

+ : 5

; i

) 

d

(

k : 5

; i

)

h

d

(

k : 5

; i

)

: (5 : 3

: 2

: 2

: 52) Using equations (5.3.2.2.48)-(5.3.2.2.51) in equation (5.3.2.2.52) leads to h d

(

k

+ : 5

; i

) = h

d

(

k : 5

; i

)(1 : : 5



dd  z

d

) + 

de  z

d

~ h

(

k

) 1 :

+ 

de  z

d 

dd  z

d

+ : 5



dd  z

d : (5 : 3

: 2

: 2

: 53) The moisture budget for the do wndraft is dev eloped in sev eral steps. First, the do wndraft w ater v apor mixing ratio before ev aporation, but after en trainmen t, is calculated. This is done using q

d

(

k; i ) = q

d

(

k : 5

; i

)(1 : : 5



dd  z

d

) + 

de  z

d

~ q

(

k

) 1 :

+ 

de  z

d 

dd  z

d

+ : 5



dd  z

d : (5 : 3

: 2

: 2

: 54) Next, equations (5.3.2.2.14) and (5.3.2.2.15) giv e the mixing ratio, q

vd , that the updraft or do wndraft w ould ha v

e if saturated. Hence, the amoun t of moisture that is necessary to k eep the do wndraft from cloud t ype i

saturated in la y

er k is q e

(

k; i ) = [ q

d

(

k; i ) q

vd (

k; i )]

: (5 : 3

: 2

: 2

: 55) Next w e c hec k whether the updraft produces enough rain to sustain saturation in the do wndraft b y requiring that X c

0

 z

(

k

)



u

(

k : 5

; i

)

q

l (

k : 5

; i

) X 

(

i

) z

(

k

)



d

(

k

+ : 5

; i

)

q

e

(

k; i )

> 0

: (5 : 3

: 2

: 2

: 56) If this is not the case, a do wndraft is not allo w

ed to exist. Ha ving dened the discretized v ersions of the equations from the static con trol, w e no w can describe the procedure. Using the larger-scale temperature and moisture elds ( T 0 ; q 0

) at time t

0

, and giv en a functional or empirical relationship for 

d

,



de , and 

dd , the equations from the static con trol are used to calculate 

ue ; h u

(

z; i )

; h d

(

z; i )

; q u

(

z; i )

; q d

(

z; i )

;  u

(

z; i ), and 

d

(

z; i ) for cloud t ype i

. These are needed to determine the total cloud w ork function A

tot using A tot (

i

) = A

u

(

i

) + A d

(

i

)

: (5 : 3

: 2

: 2

: 57) 59The discretized v

ersions of equations (5.3.2.2.40) and (5.3.2.2.42) that are used to determine the cloud w ork functions for updrafts and do wndrafts are A u

(

i

) = k

=

ktop X k =

LF C " g c p

T

(

k : 5) 

u

(

k : 5

; i

)

 h

u

(

k : 5

; i

) ~ h



(

k : 5) 1 + (

k : 5) !

 (

z

(

k 1) z

(

k

))

# (5 : 3

: 2

: 2

: 58)

and A

d

(

i

) = k

=

sur X k =

z

0 " g c p

T

(

k : 5) 

d

(

k : 5

; i

)

 h

d

(

k : 5

; i

) ~ h



(

k : 5) 1 + (

k : 5) !

 (

z

(

k

) z

(

k 1)) # : (5 : 3

: 2

: 2

: 59)

The k ernels of cloud t ype i

are b y denition the c hanges of the cloud w ork functions due to another subensem ble, i

0 . Th us, follo wing Lord (1978), T 0 and q

0 are modied b y an arbitrary amoun t of mass 
ux, m 0 b



t

0 , from the i

0

subensem ble. This is done for ev ery possible subensem ble and can be written in the sym bolic form  T 0 (

k; i ) =  T

(

k

) + 

i 0

(

 T

(

k

))

m 0 b



t

0 ; (5 : 3

: 2

: 2

: 60)
 q 0 (

k; i )= q (

k

) + 

i 0

( q (

k

))

m 0 b



t

0 : (5 : 3

: 2

: 2

: 61) The  terms, whic h are c hanges per unit m b

(

i

), are easily calculated from budget considerations as in Lord (1978). With the do wndraft terms, the moist static energy budget of la y

er k and cloud t ype i

becomes  p

(

k

) g



i 0 (

~ h

(

k; i )) = + ( 

u

(

k : 5

; i

) 

(

i

)



d

(

k : 5

; i

))

~ h

(

k : 5) ( 

u

(

k

+ : 5

; i

) 

(

i

)



d

(

k

+ : 5

; i

))

~ h

(

k

+ : 5) ( e

u

(

k; i ) + 

(

i

)

e

d

(

k; i ))

~ h

(

k

) + d

u

(

k; i )

h

u

(

k

+ : 5

; i

) + h

u

(

k : 5

; i

) 2 + 

(

i

)

d

d

(

k; i )

h

d

(

k

+ : 5

; i

) + h

d

(

k : 5

; i

) 2 ; (5 : 3

: 2

: 2

: 62)

60where e

u

(

k; i )

andd u

(

k; i ) are the en trainmen t and detrainmen t for the updraft, and  p

(

k

) is dened b y  p

(

k

) = p

(

k

+ : 5) p

(

k : 5). A simple ph ysical in terpretation of the terms on the righ thand side can be understood b y looking at Fig. 5.2. The rst term is the subsidence on top of the la y

er, the second is the subsidence on the bottom of the la y

er. This subsidence is an en vironmen tal compensatory mass 
ux due to the updraft and do wndraft mass 
uxes inside the cloud. Note that belo w z

u the \compensatory subsidence" ma y be compensatory uplifting, since in that case only do wndrafts exist. The third term represen ts en trainmen t in to the updraft and do wndraft; the fourth term represen ts detrainmen t from the edges of the updraft; the fth term represen ts detrainmen t from the edges of the do wndraft. For the moisture budget,  p

(

k

) g



i 0 (~ q (

k; i )) = + ( 

u

(

k : 5

; i

) 

(

i

)



d

(

k : 5

; i

))~ q (

k : 5) ( 

u

(

k

+ : 5

; i

) 

(

i

)



d

(

k

+ : 5

; i

))~ q (

k

+ : 5) ( e

u

(

k; i ) + 

(

i

)

e

d

(

k; i ))~ q (

k

) + d

u

(

k; i )

q

u

(

k

+ : 5

; i

) + q

u

(

k : 5

; i

) 2 + 

(

i

)

d

d

(

k; i )

q

d

(

k

+ : 5

; i

) + q

d

(

k : 5

; i

) 2 : (5 : 3

: 2

: 2

: 63)

A t the cloud top, do wndrafts ha v

e no eects and updrafts detrain all their mass.  p

(

ktop ) g



i 0 (

~ h

(

ktop; i )) = 

u

(

ktop + : 5

; i

)

~ h

(

ktop + : 5) e u

(

ktop; i )

~ h

(

ktop ) + d

u

(

ktop; i )

h

u

(

ktop + : 5

; i

) + h

u

(

ktop; i ) 2

+ 

u

(

ktop; i )

h

u

(

ktop; i ) (5 : 3

: 2

: 2

: 64) and  p

(

ktop ) g



i 0 (~ q (

ktop; i )) = 

u

(

ktop + : 5

; i

)~ q (

ktop + : 5) e u

(

ktop; i )~ q (

ktop ) + d

u

(

ktop; i )

q

u

(

ktop + : 5

; i

) + q

u

(

ktop; i ) 2

+ 

u

(

ktop; i )

q

u

(

ktop; i )

: (5 : 3

: 2

: 2

: 65) 61

# [ η(k - .5, i) -ε(i)ηdo

# (k - .5, i)]ψ(k - .5)

[e(k,i) +ε(i)edo

# (k,i)]ψ(k) [ η(k + .5,i) -ε(i)hdo

# (k+ .5,i)]ψ(k + .5) +ε(i)d do

# (k,i)ψ do

# (k - 5,i) +ψ do

# (k + 5,i) 2

+d(k,i)ψc

# (k - .5,i) +ψc

# (k + .5,i) 2


# Figure 5.2 Illustration of budget for thermodynamic variableψin layer k. k - .5

k

k + .5



# Cloud
Updraft Originating Level Downdraft Originating Level

Figure 5.3 Conceptual picture of convection parameterized in Grell scheme. 62Here  p

(

ktop ) = p

(

ktop + : 5) p

(

ktop : 5). Note that in the fourth term w e ha v

e included the detrainmen t of all the cloud mass at the cloud top. Finally , at the surface  p

(

ksur ) g



i 0 (

~ h

(

ksur ; i

)) = 

(

i

)



d

(

ksur : 5

; i

))

~ h

(

ksur : 5) + 

(

i

)



d

(

ksur ; i

)

h

d

(

ksur ; i

)  (

i

)

e

d

(

ksur : 5

; i

)

~ h

(

ksur : 5) + 

(

i

)

d

d

(

ksur ; i

)

h

d

(

ksur ; i

) + h

d

(

ksur : 5

; i

) 2 (5 : 3

: 2

: 2

: 66)

and p

(

ksur ) g



i 0 (~ q (

ksur ; i

)) = 

(

i

)



d

(

ksur : 5

; i

))~ q (

ksur : 5) + 

(

i

)



d

(

ksur ; i

)

q

d

(

ksur ; i

)  (

i

)

e

d

(

ksur : 5

; i

)~ q (

ksur : 5) + 

(

i

)

d

d

(

ksur ; i

)

q

d

(

ksur ; i

) + q

d

(

ksur : 5

; i

) 2

; (5 : 3

: 2

: 2

: 67)

with  p

(

ksur ) = p

(

ksur + : 5) p

(

ksur : 5). Here, the rst term is the compensatory en vironmen tal mass 
ux, the second term is the detrainmen t of all do wndraft air at the bottom, the third term is en trainmen t in to the do wndraft, and the fourth term is the detrainmen t of air around the do wndraft edges. The new thermodynamic elds, T 0 0 (

k; i 0 ) and q

0

0 (

k; i 0 ), are then used again from the static con trol to calculate new cloud properties and a new cloud w ork function, A

0 tot (

i

0 ; i

). Note that T

0 0

and q

0 0

are no w functions of the subensem ble i

0

. From the denition of the k ernel w e then can calculate the k ernels simply as K (

i; i 0 ) = A

0 tot (

i

0 ; i

) A

tot (

i

) m b

\ t

\ : (5 : 3

: 2

: 2

: 68) Next, w e go bac k to the original elds and modify those with the large-scale adv ectiv e c hanges to get T

\(

k

) = T 0 + 

@T @ t 

ADV 

t (5 : 3

: 2

: 2

: 69) and q

\(

k

) = q

0

+ 

@q @ t 

ADV 

t; (5 : 3

: 2

: 2

: 70) 63where (5.3.2.2.69) and (5.3.2.2.70) are applied o v

er  t

= 30 min. The double prime quan tities are then used again b y the static con trol, whic h will calculate new cloud properties, and so new cloud w ork functions, A

tot \(

i

), will be determined. Next, the large-scale forcing (b y denition the c hange of the cloud w ork function due to large-scale eects only) is calculated using F

(

i

) = A

tot \(

i

) A

tot (

i

)  t : (5 : 3

: 2

: 2

: 71) The large-scale forcing and the k ernels are then both used b y the dynamic con trol to estimate the cloud base mass 
ux distribution function, m b

, using an IMSL subroutine to solv e the linear programm ing problem. Finally , the feedbac k to the larger-scale en vironmen t is simply giv en b y  @T (

k

) @ t



CU = i 0 MAX X i 0 =1 

0 i (

T

(

k

))

m b

(

i

0 ) (5 : 3

: 2

: 2

: 72) and 

@q (

k

) @ t



CU = i 0 MAX X i 0 =1 

0 i (

q

(

k

))

m b

(

i

0 )

; (5 : 3

: 2

: 2

: 73) where the precipitation can be calculated using P =

i 0 MAX X i 0 =1 k

=

ktop X k =1 c

0

 z

(

k

)

q

l (

k

+ : 5

; i

)

m u

(

k

+ : 5

; i

) i 0 MAX X i 0 =1 k

=

ktop X k =1  z

(

k

)

q

ev (

k

+ : 5

; i

)

m d

(

k

+ : 5

; i

)

: (5 : 3

: 2

: 2

: 74)

5.3.2.3 The Grell sc heme This is a v ery simple sc heme that w as constructed to a v

oid rst-order sources of errors (Grell 1993). The v ery simplistic conceptual picture of ho w this parameterization is en visioned to function is sho wn in Fig. 5.3. Clouds are pictured as t w

o steady-state circulations, caused b y an updraft and a do wndraft. There is no direct mixing bet w

een cloudy air and en vironmen tal air, except at the top and the bottom of the circulations. The cloud model that is used to calculate cloud properties in this sc heme is form ulated 64with only a few equations. Mass 
ux is constan t with heigh t, and there is no en trainmen t or detrainmen t along the cloud edges. We can simply write

m u

(

z

) = m u

(

z

b

) = m b (5 : 3

: 2

: 3

: 1) and m d

(

z

) = m d

(

z

0

) = m 0 (5 : 3

: 2

: 3

: 2) for the mass 
ux of the updraft ( m u

) and the do wndraft ( m d

). Here m b

and m 0

are simply the mass 
uxes of the updraft and do wndraft at their originating lev el. If it is assumed that the conditions at originating lev els are giv en b y the en vironmen t, for an y thermodynamic v ariable , the budget inside the cloud simply becomes

u (

z

)=~ (

z

b

) + S

u

(

z

)

; (5 : 3

: 2

: 3

: 3) and d

(

z

)=~ (

z

0

) + S

d

(

z

)

; (5 : 3

: 2

: 3

: 4) where is a thermodynamic v ariable, the tilde denotes an en vironmen tal v alue, and S stands for sources or sinks. For moist static energy ~ h (

z

) = Cp ~ T

(

z

) + gz + L

~ q

(

z

)

; (5 : 3

: 2

: 3

: 5) equations (3) and (4) simply become h

u

(

z

) = ~ h

(

z

b

) (5 : 3

: 2

: 3

: 6) and h

d

(

z

) = ~ h

(

z

0

)

: (5 : 3

: 2

: 3

: 7) For the moisture budget of the updraft w e can mak e use of the appro ximate equations (5.3.2.2.14) and (5.3.2.2.15) to calculate the mixing ratio inside the cloud if saturation is assumed. T ogether with equations (5.3.2.3.3) and (5.3.2.3.4), this will giv e us S

u and S

d

, the condensation and ev aporation. Note also that no cloud w ater is assumed to exist; all w ater is con v

erted to rain. 65Giv en boundary conditions, equations (5.3.2.3.1)-(5.3.2.3.7) ha v

e t w

o unkno wns, m b

, and m 0

. In order to lea v

e only one unkno wn v ariable, the originating mass 
ux of the do wndraft is made a function of the updraft mass 
ux and the reev aporation of con v

ectiv e condensate, as in the previous section (see equations (5.3.2.2.20)-(5.3.2.2.24)). Therefore,

m 0

= I1 m b I2

= m b

: (5 : 3

: 2

: 3

: 8) Here, 1 is the precipitation eciency . To specify boundary conditions, w e assume h u

(

z

) = h

u

(

z

b

) = MAX (

~ h

(

z

))

; (5 : 3

: 2

: 3

: 9) with z

 z

b

; and h u

(

z

T

) = ~ h



(

z

T

)

; (5 : 3

: 2

: 3

: 10) where the asterisk denotes a saturation v alue. Similarly , for the do wndraft, h d

(

z

) = h

d

(

z

0

) = MIN (

~ h

(

z

))

: (5 : 3

: 2

: 3

: 11) Ph ysically , for both, updraft and do wndraft, w e allo w for maxim um buo y

ancy . For this deep con v

ection sc heme, the cloud base for the updraft is not restricted to the boundary la y

er, but can be an ywhere in the troposphere. F eedbac k to the larger-scale equations T o a v

oid zero-order sources of errors, the feedbac k m ust include the cooling eects of moist con v

ectiv e do wndrafts. Furthermor e, lateral mixing should nev er be excessiv e, especially if the cloud properties ha v

e been calculated with a steady-state cloud model. Keeping in mind the conceptual picture in Fig. 5.3, the feedbac k for this sc heme is en tirely determined b y compensating mass 
uxes and detrainmen t at cloud top and bottom. Conceptually , no a v

eraging (suc h as the normally used top-hat or Reynolds a v

eraging methods) is necessary . This does not mean that scale-separation is not required, but for this parameterization it is not necessary to assume that the fractional area co v

erage is v ery small. Note, ho w

ev er, that an y parameterization can only mak e sense if a clear scale separation exists. None of the parameterized eects ma y be resolv ed b y the larger-scale. 66Assuming that the conceptual picture in Fig. 5.3 happens in only one grid bo x, w e can express the c hanges caused b y the con v

ection as @ ~ h

(

k

) @ t

! CU = @h u

(

z

)

m b @z @

~ h

(

z

)

m b @z @h d

(

z

)

m 0 @z

+ @

~ h

(

z

)

m 0 @z (5 : 3

: 2

: 3

: 12) and 

@q (

k

) @ t



CU = @q u

(

z

)

m b @z @

~ q

(

z

)

m b @z @q d

(

z

)

m 0 @z

+ @

~ q

(

z

)

m 0 @z

: (5 : 3

: 2

: 3

: 13) Because of the simplicit y of the static con trol, these equations can be further simplied to giv e @

~ h

(

k

) @ t

!

CU = m b

@

~ h

(

z

) @z

(1 

) + m b

(

@h u

(

z

) @z 

@h d

(

z

) @z

) (5 : 3

: 2

: 3

: 14)  @q (

k

) @ t



CU = m b

@

~ q

(

z

) @z

(1 

) + m b

(

@q u

(

z

) @z 

@q d

(

z

) @z

)

: (5 : 3

: 2

: 3

: 15) The rainfall is dened as R  I1 m b

(1 )

: (5 : 3

: 2

: 3

: 16) The second term on the righ thand sides of equations (5.2.2.3.14) and (5.2.2.3.15) are due to do wndrafts and are zero abo v

e the do wndraft originating lev el. Belo w the updraft-air originating lev el, the rst term of the righ t-hand sides are zero and only do wndrafts aect the larger-scale en vironmen t. All v ariables in the 
ux terms can be determined from the equations of the static con trol, except m b

. Dynamic con trol Because of the simplicit y of the abo v

e equations, man y closure assumptions can be used. The most simple closure is a Kuo-t ype assumption, whic h relates the rainfall rate to the moisture con v

ergence. Ho w

ev er, more applicable seems to be a stabilit y closure. Again w e ha v

e t w

o c hoices. We could assume that the clouds will remo v

e the a v

ailable buo y

an t energy as in other mesoscale parameterizations, or that the clouds will stabilize the en vironmen t as fast as the larger-scale (or also sub-grid scale) destabilizes it, or ev en a mixture of both. Although both assumptions are easily implemen ted, w e c hose the closure whic h depends on the rate of destabilization. In this closure the c hange of the a v

ailable 67buo y

an t energy due to con v

ection osets the c hanges due to other eects (larger-scale destabilization as w ell as sub-grid scale destabilization), yielding  dABE dt 

OTH = 

dABE dt 

CU : (5 : 3

: 2

: 3

: 17) Next, the c hange due to the con v

ection is normalized in terms of the mass 
ux to read  dABE dt 

CU  m b



dABE dt 

NCU ; (5 : 3

: 2

: 3

: 18) where subscript NCU denotes the c hange of the a v

ailable buo y

an t energy due to a cloud normalized b y the cloud-base mass 
ux. Equations (5.3.2.3.17) and (5.3.2.3.18) are used to calculate m b

. The Procedure This section describes in detail the procedure necessary to calculate the con v

ectiv e feedbac k. First, w e will explain the v ery simplistic approac h to calculate a normalized feedbac k, then w e will describe ho w the closure assumption determines the mass 
ux. Using the larger-scale temperature and moisture elds (

T 0 ; q 0

) at time t

0

, h u

(

z

)

; h d

(

z

)

; q u

(

z

)

; q d

(

z

) are simply arriv ed at (see equations (5.3.2.3.6)-(5.3.2.3.10). The rst calculation is the determination of the in tegrals I

1 and I2 (calculated as residuals using equations (5.3.2.3.8) and (5.3.2.3.9). The next step is then to estimate the con v

ectiv e c hanges per unit mass 
ux (before kno wing the actual m b

's). This is done b y estimating the net c hange of a thermodynamic v ariable in a la y

er k b

y using  p

(

k

) g



( ~ (

k

)) = (1 

)( ~ (

k : 5) ~

(

k

+ : 5)) ; (5 : 3

: 2

: 3

: 19) where  p

(

k

) is dened b y  p

(

k

) = p

(

k

+ : 5) p

(

k : 5). This subsidence is an en vironmen tal compensatory mass 
ux due to the updraft and do wndraft mass 
uxes inside the cloud. Note that belo w z

u the \compensatory subsidence" ma y be compensatory uplifting, since in that case only do wndrafts exist. A t the cloud top,  p

(

ktop ) g



( ~ (

ktop )) = ~

(

ktop : 5) + u

(

ktop )

: (5 : 3

: 2

: 3

: 20) 68Here  p

(

ktop ) = p

(

ktop + : 5) p

(

ktop : 5). Finally , at the surface (the do wndraft tops)  p

(

ksur ) g



( ~ (

ksur )) = 

( ~

(

ksur : 5) + d

(

ksur ))

; (5 : 3

: 2

: 3

: 21) with  p

(

ksur ) = p

(

ksur + : 5) p

(

ksur : 5). Here, the rst term is the compensatory en vironmen tal mass 
ux, and the second term is the detrainmen t of all do wndraft air at the bottom. These normalized c hanges are also used in the calculation of the nal feedbac k (after m b

is determined), whic h is simply giv en b y  @ (

k

) @ t



CU = 

( (

k

))

m b

: (5 : 3

: 2

: 3

: 22) To calculate the mass 
ux m b

, w e dene the buo y

an t energy whic h is a v

ailable to a cloud (updraft and do wndraft) as ABE = k

=

ktop X k =

LF C " g c p

T

(

k : 5)  ~ h

(

kb ) ~ h



(

k : 5) 1 + (

k : 5) ! 

(

z

(

k 1) z

(

k

))

#
+ k

=

sur X k =

z

0 " g c p

T

(

k : 5)  ~ h

(

k

0) ~ h



(

k : 5) 1 + (

k : 5) ! 

(

z

(

k

) z

(

k 1)) # :

(5 : 3

: 2

: 3

: 23)
where is dened in equation (5.3.2.2.15). We can calculate ABE (similar to Lord 1982) for the unc hanged en vironmen t as w ell as for the en vironmen t whic h has been modied b y some arbitrary mass 
ux m 0 b



t

0 . Hence, w e can write NA = 

dABE dt 

NCU = ABE 0 ABE m 0 b



t

0 : (5 : 3

: 2

: 3

: 24) ABE are calculated using T 0 and q

0

, while ABE 0

are calculated after modication of the thermodynam ic v ariables b y an arbitrary amoun t of mass 
ux, m 0 b



t

0 , where

0 (

k

) = (

k

) + 

( (

k

))

m 0 b



t

0 : (5 : 3

: 2

: 3

: 25) For a closure whic h depends on the rate of destabilization, w e ha v

e to calculate the c hange in the a v

ailable buo y

an t energy due to large-scale or other subgrid-scale eects. We modify the thermodynam ic elds with \( k

) = 0

+ 

@ @ t 

LS +

SUBG 

t; (5 : 3

: 2

: 3

: 26) 69where (5.3.2.3.26) is applied at ev ery timestep  t

. These double prime quan tities are then used to calculate the c hanges in the a v

ailable buo y

an t energy due to \non-con v

ectiv e" eects. As a result, the equation for the mass 
ux becomes

m b

= ABE \ ABE ( ABE 0 ABE )

m 0 b

: (5 : 3

: 2

: 3

: 27) 5.3.3 P aram eterization of shallo w con v

ection The shallo w con v

ection sc heme is constructed to be able to serv e t w

o tasks. It parameterizes planetary boundary la y

er (PBL) forced shallo w non-precipitating con v

ection as w ell as mid-tropospheric shallo w con v

ection caused b y other sub-grid scale eects (suc h as cloud top radiational cooling). The rst migh t not be necessary when the parameterization is coupled to a higher order closure PBL sc heme. It will transport moisture from inside the boundary la y

er in to the la y

ers just abo v

e the boundary la y

er. This is accomplished b y em ulating bubbles (forced b y surface heat and moisture 
uxes only , with strong lateral mixing) whic h rise without precipitation formation through the top of the boundary la y

er in to the free atmosphere, where they then lose their buo y

ancy . Because of the strong lateral mixing, they usually do not rise more than 50-75 m b. The ph ysics in v

olv ed in describing the second kind of shallo w con v

ection is the same, except for the forcing. T o parameterize this t ype of con v

ection w e assume that a \con v

ectiv e elemen t" can be c haracterized b y a bubble whic h rises through sev eral model la y

ers. It is assumed to be forced b y planetary boundary la y

er 
uxes or radiational cooling tendencies. Some of the elemen ts of this parameterization are based on an Arak a

w

a-Sc h

ubert t ype sc heme (section 5.3.2.2) and some are based on the simple one-cloud sc heme described in section 5.3.2.3. Ho w

ev er, the clouds (shallo w \con v

ectiv e elemen ts\) are c haracterized b y dieren t properties. They usually ha v

e large mixing, are non-precipitating, and ha v

e no con v

ectivescale do wndrafts. They are forced b y subgrid-scale processes only . The follo wing description will be focused on dierences from the previously described models. Since the sole purpose of this sc heme is to represen t \v ery" shallo w con v

ection, it is also constructed as a one-cloud sc heme. Although it implicitly uses equations (5.3.2.2.1)- (5.3.2.2.4), considerable simplications can be made b y assuming strong lateral mixing 70(detrainmen t being equally as strong as en trainmen t). Equations (5.3.2.2.1) through (5.3.2.2.4) then read  = 0 ; (5 : 3

: 3

: 1)
 e

= 

d

= : 2 r

; (5 : 3

: 3

: 2) and @ c @z

= : 2 r

( ~ c

) + S

c

; (5 : 3

: 3

: 3) where r in equation (5.3.3.2) is the radius of the elemen t. The parameterization will be sensitiv e to the c hoice of r

. For this t ype of con v

ection w e assume r

= 50 m . When assuming that no precipitation forms or ev aporates, equations (5.3.3.1)-(5.3.3.3), together with initial conditions (5.3.2.2.25) and (5.3.2.2.26), form a simple set of equations to determine the properties of the con v

ectiv e elemen t, if r is giv en. Without precipitation formation, S

c in equation (5.3.3.3) is zero. F or the feedbac k, equations (5.3.2.2.32)-(5.3.2.2.34) simply become F S s

(

z

)

 [ s

c

(

z

)  s

(

z

)]

m c

; (5 : 3

: 3

: 4) F S q

(

z

)

 [ q

c

(

z

)  q

(

z

)]

m c

; (5 : 3

: 3

: 5) F S l (

z

)

 l

(

z

)

m c

= 0 : (5 : 3

: 3

: 6) The only unkno wn in these equations is the mass 
ux. It is determined in the dynamic con trol, where w e mak e use of the denition of the cloud w ork function (5.3.2.2.40) and simply impose 

dA (

scl ) dt



CU = 

dA (

scl ) dt



SUBG : (5 : 3

: 3

: 7) Note that since the cloud w ork function is independen t of mass 
ux (mass 
ux is constan t with heigh t), equation (5.3.2.2.40) for cloud-t ype scl simplies to A (

scl ) = Z z

T
z B g Cp T

(

z

) 1 1 + (

h

c

(

z

) ~ h



(

z

))

dz : (5 : 3

: 3

: 8) Subscript CU refers to the eects due to con v

ection, and SUBG to eects due to sub-grid scale forcing. A

(

scl ) becomes simply the buo y

ancy whic h is a v

ailable for that particular 71cloud scl . Therefore, ph ysically , the c hange of the eciency of kinetic energy generation due to cloud scl is directly proportional to the buo y

ancy generation b y sub-grid scale forcing. T o arriv e at a useful closure, the term on the left hand side of equation (5.3.3.7) is normalized b y the mass
ux to yield

m c



dA (

scl ) dt



NCU = 

dA (

scl ) dt



SUBG : (5 : 3

: 3

: 10) Here, the subscript NCU no w stands for the c hange of A due to a unit mass of cloud scl . The v ariables in equation (5.3.3.10) are kno wn, except for m c

. After using (5.3.3.10) to calculate m c

, w e can then calculate the feedbac k. Note that in equation (5.3.3.2), mc is not dependen t on heigh t, and is simply the cloud base mass 
ux. It should be noted here that the abo v

e described parameterization will greatly benet from a high v ertical resolution. In some instances it ma y be of use to allo w the shallo w con v

ection sc heme to be called sev eral times in a column (stac k

ed on top of eac h other), since dieren t sub-grid-scale forcing mec hanism s ma y act at the same time in one column, but at dieren t lev els.

725.4 Planetary boundary lay er param eterizations 5.4.1 Surface-Energy equation Ov er land, the surface temperature T g is computed from a surface energy budget that is base on the \force-restore" method dev eloped b y Blac k

adar (Zhang and An thes 1982). The budget equation is C

g

@T g @ t = R

n H

m H s L

v

E

s

; (5 : 4

: 1

: 1) where C

g

is the thermal capacit y of the slab per unit area, R

n

is the net radiation, H

m is the heat 
o w in to the substrate, H s

is the sensible heat 
ux in to the atmosphere, L

v

is the laten t heat of v aporization, and E

s

is the surface moisture 
ux. Blac k

adar (1979) sho ws that the follo wing form ulation enables the amplitude and phase of the slab temperature to be iden tical to the surface temperature of a real soil la y

er of uniform thermal conductivit y  and heat capacit y per unit v olume C

s

, with C

g related to these parameters and the angular v elocit y of the earth 
 b y C

g

= : 95 

C s 2



1

=

2

: (5 : 4

: 1

: 2) The thermal capacit y

,

C

g

, is related to a parameter called the thermal inertia, 

, where  is  = ( C s

)

1

=

2

: (5 : 4

: 1

: 3) From (5.4.1.2) and (5.4.1.3), C

g

= 3 : 29310 6

; (5 : 4

: 1

: 4) where  (

cal cm 2

K 1

s

1 2

; 1 cal = 4 : 18 J

) is specied in the model as a function of land-use c haracteristic (Appendix 4). The terms on the righ t hand side of (5.4.1.1) are described as follo ws:

735.4.1.1 Net radiativ e 
ux R

n Radiation is the driving force of the diabatic planetary boundary la y

er (PBL) and is the most importan t componen t of the slab-energy budget. R n = Q

s

+ Is (5 : 4

: 1

: 5) where Q

s

and Is are the net surface short w

a

v

e and longw a

v

e irradiances. a. Clear Sky For clear sky , the amoun t of solar radiation absorbed b y the slab, including m ultiple re
ection of short w a

v

es, is appro ximated as Q s

= S

0

(1 A

)

 cos ; (5 : 4

: 1

: 6) where S

0

is the solar constan t (1395.6 W m 2

), A is the albedo. is the zenith angle, and  is the short-w a

v

e transmissivit y

. The term cos is giv en b y cos = sinsin + coscoscosh 0

; (5 : 4

: 1

: 7) where  represen ts the latitude of the location,  the solar declination, and h

0

the local hour angle of the sun (Sellers. 1974). The short-w a

v

e transmissivit y for m ultiple re
ection (Benjamin 1983) is
 = a [ 

s

+ (1 s )(1 b

)] (1 X R

A

) ; (5 : 4

: 1

: 8) where a is the absorption transmissivit y

,

s is the scattering transmissivit y

,

b is the bac kscattering coecien t, and X R is the m ultiple re
ection factor X R = ad (1 sd )

b

d

; (5 : 4

: 1

: 9) where the subscript d denotes diuse. All the clear-air transmissivities ( a ; s ; ad ;  sd ) and bac kscattering coecien ts ( b

and b d

) are determined as a function of path length and precipitable w ater from a look-up table from the Carlson and Boland (1978) radiativ e transfer model. Transmissivities are then adjusted for surface pressure as follo ws: 

= 1+( 

0 1)

p

s 1013 : 25 ; (5 : 4

: 1

: 10) 74where 

0

is the transmissivit y from the look-up table (appendix 2) obtained b y assuming the surface pressure is 1013.25m b, and p

s

is the surface pressure at the location. The net longw a

v

e radiation, Is , is equal to the sum of the outgoing ( I "

) and do wn w

ard ( I #

) longw a

v

e radiation. The outgoing longw a

v

e radiation is I "

= 

g



SB T

4 g

; (5 : 4

: 1

: 11) where 

g is the slab emissivit y

, T g is the ground temperature, and 

SBthe StefanBoltzmann constan t. The do wn w

ard longw a

v

e radiation absorbed at the surface is I #

= 

g



a



SB T

4 a

; (5 : 4

: 1

: 12) where T a is the atmospheric temperature in the la y

er abo v

e the surface, and 

a

, the atmospheric longw a

v

e emissivit y

, is giv en b y
 a

= : 725 + : 17 log 10

w

p

; (5 : 4

: 1

: 13) in whic h w

p

is the precipitable w ater in cen timeters b. Cloudy skies F or cloudy skies, a cloud parameterization sc heme (Benjamin 1983) is used to sim ulate the eects of clouds on short-w a

v

e and do wn w

ard longw a

v

e radiation. Groups of sigma lev els are c hosen to correspond to lo w-, middle-, and upper-cloud la y

ers based upon an assumed surface pressure of 1000m b. The clouds belo w 800m b are designated as lo w clouds, middle clouds are those bet w

een 800m b and 450m b, and upper clouds are those abo v

e 450m b. The atten uation of short-w a

v

e radiation b y cloud is parameterized with absorption ( ac ) and scattering ( sc ) transmissivities. The transmissivities through the three cloud la y

ers are giv en b y ac = 3 Y i =1 [1 (1 ai )]

n

i (5 : 4

: 1

: 14) and sc = 3 Y i =1 [1 (1 si )]

n

i

; (5 : 4

: 1

: 15) 75where i = 1 ;

2

;

3 represen ts lo w, middle, and high clouds, respectiv ely ,

n

i

is the cloud fraction, and ai and si are giv en in table 5.1. The minim um short-w a

v

e absorption transmissivit y is set at 0.7, and the minim um scattering transmissivit y is set at 0.44. The cloud fraction is based on relativ e h umidit y

. Cloud fraction at lo w and middle lev els is n = 4 : 0

RH 3

: 0

; (5 : 4

: 1

: 16) and in the upper atmosphere n = 42 : 5

RH 1

: 5

; (5 : 4

: 1

: 17) where RH is the maxim um relativ e h umidit y found in the model la y

ers within the lo w, middle, or upper cloud la y

ers. The expression for eectiv e short-w a

v

e transmissivit y under cloudy skies is  = ac sc a [ s + (1 s )(1 b

)] (1 X c

A

) ; (5 : 4

: 1

: 18) where the m ultiple re
ection factor for cloudy skies ( X c

) is dened as X c

= ad ac (1 sd sc )

 b

d (5 : 4

: 1

: 19) in whic h  b

d

, the mean bac kscattering coecien t, is

 b d

= b

d

(1 sd ) + (1 

sc

) (1 sd ) + (1 sc ) : (5 : 4

: 1

: 20) The cloud enhancemen t of long-w a

v

e radiation inciden t on the ground is expressed as I #

0 = I

# 1 + 3 X i =1 c

i n

i ! ; (5 : 4

: 1

: 21) where c

i

are the enhancemen t coecien ts at dieren t lev els (table 5.2).

76

# Table 5.1 Cloud absorption and scattering transmissivities. Cloud Level Absorption (τai

# ) Scattering (τsi

# ) Low 0.80 0.48 Middle 0.85 0.60

High 0.98 0.80

Table 5.2 Enhancement coefficientsci

# on longwave radiation due to clouds. Cloud Level Coefficient Low 0.26 Middle 0.22

High 0.06

775.4.1.2 Heat Flo w in to the Substrate H

m The transfer of heat due to molecular conduction is calculated from the equation H m = K m

C

g

(

T g T m )

; (5 : 4

: 1

: 22) where K m is the heat transfer coecien t expressed as K m = 1 : 18
, 
 is the angular v elocit y of the earth, and T m is the temperature of the substrate, whic h is presen tly tak en to be a constan t v alue equal to the mean surface-air temperature o v

er the period of sim ulation. If the model is used in a forecast mode rather than a researc h mode, T m ma y be set equal to the mean surface temperature of the previous da y

. 5.4.1.3 Sensible-Heat Flux H s

and Surface Moisture Flux E

s These 
uxes are computed in dieren t w a

ys, depending upon what PBL parameterization is used. Details will be described in the next sections. 5.4.2 Bulk-aerodynam ic param eterization The bulk-aerodynamic option of the PBL ph ysics follo ws Deardor (1972). It is a v ery inexpensiv e c hoice. The surface-heat 
uxes are giv en b y H s

= 

a

c

pm C



C

u

(



g 

a

)

V ; (5 : 4

: 2

: 1) where 

a

and 

a

are densit y and poten tial temperature at the lo w

est model la y

er, C



and C u are exc hange coecien ts (Deardor 1972) dened as C u

= C

uN 

1 R

iB R iC  (5 : 4

: 2

: 2) and C



= C

N 

1 R

iB R iC  (5 : 4

: 2

: 3) for stable conditions (0  R

iB  : 9

R

ic

), and C u

= 1 1 C uN 25 exp (

: 26 : 03 2

) (5 : 4

: 2

: 4) and C



= 1 1 C N + 1 C u 1 C uN (5 : 4

: 2

: 5) 78for the unstable case ( R

iB  0). Here C

uN and C

N are the neutral v alues for C

u

and C



, and are giv en b y C

uN =  k 1

ln 

: 025 h z 0  + 8 : 4

 1 (5 : 4

: 2

: 6) and C

N = 

0

: 74 k 1

ln 

: 025 h z 0  + 7 : 3

 1

; (5 : 4

: 2

: 7) where R

ic = 3 : 05, h is the depth of the lo w

est model la y

er, is dened as

= log 10 ( R

iB ) 3

: 5

; (5 : 4

: 2

: 8) and the v elocit y V is giv en b y V = ( V

2 a

+ V

2 c

)

1

=

2

: (5 : 4

: 2

: 9) V a is the wind-speed at the lo w

est model la y

er, and V c is a con v

ectiv e v elocit y

, whic h is importan t under conditions of lo w mean wind-speed and is dened under unstable and neutral conditions as V c = 2( 

g 

a

)

1

=

2

; (5 : 4

: 2

: 10) while it is zero under stable conditions. The surface moisture 
ux is E s

= 

a

C



C

u

M (

q

vs (

T g ) q

va )

V ; (5 : 4

: 2

: 11) where M is the moisture a v

ailabilit y parameter whic h v aries from 1.0 for a w et surface to 0.0 for a surface with no poten tial for ev aporation. The moisture a v

ailabilit y is specied as a function of land-use category (Appendix 4). The model results are often quite sensitiv e to the v alue used for M . The surface momen tum 
ux is giv en b y
s = 

a

C

D

V

2

; (5 : 4

: 2

: 12) where the drag coecien t

C

D is dened as C D = C

0 D + 310 3

 

s  s

+ 9800 

: (5 : 4

: 2

: 13) 79The second term in (5.4.2.13), in v

olving the surface geopoten tial 

s

, is a correction for elev ated terrain (Blec k,1977). The expression for C

0 D follo ws Deardor (1972), where C 0 D = C

2 u (5 : 4

: 2

: 14) 5.4.3 Blac k

adar High-resolution model A revised v ersion of Blac k

adar's PBL model (Blac k

adar, 1976, 1979; Zhang and An thes, 1982) is used to forecast the v ertical mixing of horizon tal wind ( u and v

), poten tial temperature ( 

), mixing ratio ( q

v

), cloud w ater ( q

c

), and ice ( q

i

). The surface heat and moisture 
uxes are computed from similarit y theory . First the friction v elocit y

,

u



, is computed based on u



= MAX kV ln z

a z 0 m ; u 

0

! ; (5 : 4

: 3

: 1) where u



0

is a bac kground v alue (0.1 ms 1

o

v

er land and zero o v

er w ater) and V is giv en b y (5.4.2.9). The surface-heat 
ux is computed from H s

= Cpm 

a

ku 

T  ; (5 : 4

: 3

: 2) where T  = 

a 

g ln z

a z 0 h

; (5 : 4

: 3

: 3) where z

0

is the roughness parameter, z

a is the heigh t of the lo w

est 

-lev el, and m and h are nondimensional stabilit y parameters that are a function of the bulk Ric hardson n um ber R

iB , whic h is giv en b y R

iB = gz a  a 

va 

vg V 2 ; (5 : 4

: 3

: 4) where the subscript v represen ts virtual poten tial temperature. There are four cases possible: a. Stable case F or the stable case, R

iB > R ic

, where the critical Ric hardson n um ber R

i c

is dened as R

ic = : 2

: (5 : 4

: 3

: 5) 80In this case, u



= u



0

; (5 : 4

: 3

: 6)

m = h

= 10 ln z

a z 0

; (5 : 4

: 3

: 7) and H s

= M ax ( 250 W m 2

; c

pm 

a

ku 

T  )

: (5 : 4

: 3

: 8) b. Mec hanically driv en turbulence For this case 0  R

iB  R

ic

, and w e get

m = h

= 5

 R

iB 1 : 1 5

R

iB  ln z

a z 0

: (5 : 4

: 3

: 9) c. Unstable (forced con v

ection) Here R

iB < 0 and j

h=L j  1

: 5, where the Monin-Obukho v length, L

, is dened as L = c

pm 

a



a

u

3  kgH s (5 : 4

: 3

: 10) and h is the heigh t of the PBL. In this case, m = h

= 0, and z

a

=L = R

iB ln z

a z 0

. d. Unstable (free con v

ection) Here R

iB < 0 and j

h=L j

> 1

: 5. In this case
h = 3

: 23 

z

a L

 1

: 99 

z

a L



2 0

: 474 

z

a L



3

; (5 : 4

: 3

: 11) and

m = 1

: 86 

z

a L

 1

: 07 

z

a L



2 0

: 249 

z

a L



3

: (5 : 4

: 3

: 12) where z

a

=L is restricted to be no less than -2.0 in this appro ximation. For z

a

=L equal to -2.0, h

= 2 : 29, and m = 1 : 43. In the general case, z

a

=L is a function of m and (5.4.3.12) is an implicit equation requiring an iterativ e solution. To sa v

e time, w e appro ximate z

a

=L as an explicit function of R

iB , suc h that z

a L = R

iB ln z

a z 0

: (5 : 4

: 3

: 13) 81The abo v

e sc heme ensures con tin uit y of m for all v alues of R

iB . The form ulation for the surface moisture 
ux in the m ulti-la y

er case w as deriv ed from Carlson and Boland (1978), where E

s

= M  a

I 1

(

q

vs (

T g ) q

va )

; (5 : 4

: 3

: 14) and I 1

= ku 

 ln 

ku 

z

a K a + z

a z l

 h

 1

: (5 : 4

: 3

: 15) The quan tit y z

l

is the depth of the molecular la y

er (0.01 m o v

er land and z

0

o

v

er w ater) and K a

is a bac kground molecular diusivit y equal to 2 : 410 5

m 2

s 1

. Ov er land, the roughness length z

0 is specied as a function of land-use category (Appendix 4). Ov er w ater, z

0

is calculated as a function of friction v elocit y (Delsol et al. , 1971) suc h that z

0

= 0 : 032 u

2 

=g + z

0

c

; (5 : 4

: 3

: 16) where z

0

c

is a bac kground v alue of 10 4

m . The Blac k

adar sc heme considers t w

o dieren t PBL regimes, the nocturnal regime and the free-con v

ection regime. The rst three cases (stable, mec hanically driv en turbulence, and forced con v

ection) are in the nocturnal regime, whic h is usually stable or at most marginally unstable. Nocturnal Regime The rst-order closure approac h is used to predict model v ariables. The ground stress is calculated from s = u 2 

; (5 : 4

: 3

: 17) where u



is computed from (5.4.3.1). The componen ts of 

s

in the x and y directions are
sx = u V a s (5 : 4

: 3

: 18) and sy = v V a s ; (5 : 4

: 3

: 19) where V a is the wind speed at the lo w

est model lev el. For surface la y

er v ariables, the prognostic equations are @ a @ t = (

H 1 H s

) ( 

a

c

pm z

1

) ; (5 : 4

: 3

: 20) 82@q va @ t = (

E

1 E

s

) ( 

a

z

1

) ; (5 : 4

: 3

: 21) @u a @ t = (

1 x sx ) ( 

a

z

1

) ; (5 : 4

: 3

: 22) @v a @ t = (

1 y sy ) ( 

a

z

1

) ; (5 : 4

: 3

: 23) and @q ca @ t = F 1 ( 

a

z

1

)

; (5 : 4

: 3

: 24) where H s

is the surface heat 
ux computed from (5.4.3.2), E

S

is the surface moisture 
ux computed from (5.4.3.14), subscript a

refers to surface la y

er v ariables, subscript 1 refers to the 
uxes at the top of the surface la y

er (Fig. 5.4), and z

1

is the heigh t of the lo w

est model la y

er. The 
uxes at the full  lev els are computed from K -theory , as described in section (5.4.4). The prognostic v ariables abo v

e the surface la y

er are computed from K -theory and an implicit diusion sc heme (Ric h

tmey er, 1957; Zhang and An thes, 1982). Free-Con v

ection Regime During strong heating from belo w, large surface heat 
uxes and a super- adiabatic la y

er occur in the lo w

er troposphere. As the buo y

an t plumes of hot air rise under suc h unstable conditions, mixing of heat, momen tum , and moisture tak e place at eac h lev el. The v ertical mixing in this sc heme is not determined b y local gradien ts, but b y the thermal structure of the whole mixed la y

er. In the Blac k

adar PBL model, the v ertical mixing is visualized as taking place bet w

een the lo w

est la y

er and eac h la y

er in the mixed la y

er, instead of bet w

een adjacen t la y

ers as in K-theory . In the surface la y

er, the prognostic v ariables are solv ed b y the analytic solution

 +1 a =  1 a

+ 

F s z

1  mh 2 F s  mh + F 1  mh 

exp   mh 

t z 1  1



+ F s 

t h

; (5 : 4

: 3

: 25) where represen ts an y prognostic v ariable, F s is the surface 
ux, F 1 is the 
ux at the top of the surface la y

er, h is the heigh t of the PBL,  t

is the time-step, and the mixing coecien t is  m = H 1

"



a

c

pm (1 

)

Z h
z 1

[ 

va 

v

(

z

0 )]

dz 0 # 1

: (5 : 4

: 3

: 26) 83Here 

is the en trainmen t coecien t (0.2) and H 1

is the heat 
ux at the top of the surface la y

er computed b y the Priestly equation H 1

= 

a

c

pm z

1

(



va 

v

1

1 2

)

3 2  2

g 27 

va  1 2 1 z 1 h z 1 3 1 (2 z

1

1 2

) 1 3

i 3 2

; (5 : 4

: 3

: 27) where z

1 is the depth of the surface la y

er and the subscript 1 1 2 refers to the second prediction la y

er abo v

e the surface (Fig. 5.4). For the v ariables abo v

e the surface la y

er, the prognostic equation is @ i @ t

=  m ( a i

)

; = ; q v

; or q c (5 : 4

: 3

: 28) @ i @ t

= w  m

( a i

)

; = u; v: (5 : 4

: 3

: 29) The v ariable w is a w eigh ting function for reducing mixing near the top of the mixed la y

er, where w = 1 z h

: (5 : 4

: 3

: 30) Care m ust be tak en at the la y

er where the top of the mixed la y

er is located because the top of the mixed la y

er does not necessarily coincide with a model lev el. 5.4.4 Vertical diusion Abo v

e the mixed la y

er, K-theory is used to predict the v ertical diusion of the prognostic v ariables, suc h that F V = p



@ @z K z

@ @z ; (5 : 4

: 4

: 1) where the eddy diusivit y

,

K z

, is a function of the local Ric hardson n um ber R

i

. Specically , K z

= K z

0

+ l 2

S

1 2 R

ic R

i R ic for R

i

< R ic (5 : 4

: 4

: 2) K z

= K z

0

; for R

i

 R

ic (5 : 4

: 4

: 3) where K z

0 = 1 m 2 s 1 ; l = 40 m ; andR ic is a critical Ric hardson n um ber whic h is a function of la y

er thic kness (m) and is dened as R ic = : 257  z

: 175
: (5 : 4

: 4

: 4) 84

# Z1 

# Z5 

# Z4 1/2 

# Z4


# Z3 

# Z3 1/2



# Z2 

# Z2 1/2


# Z1 1/2


# Za 

# N

P F

F
F
F
F
Fs 

# α

α

α

α

α θva 

# θvg 

# θ h x

x

x

x

x

Figure 5.4 Illustration of vertical grid structure for high-resolution (Blackadar) model. The top of the surface layer isz1

# ;θvg

# andθva

# are the virtual potential temperatures of the ground surface and lowest model level, respectively;PandN denote the positive and negative areas associated with a parcel of air originating at za

# and rising toh, the top of the PBL.

85According to (5.4.4.4), R

ic v

aries from 0.58 for  z = 100 m to 0.86 for  z = 1000 m . The Ric hardson n um ber is R

i

= g S @ @z (5 : 4

: 4

: 5) and S is S = 

@u @z 

2

+ 

@v @z 

2

+ 10 9

: (5 : 4

: 4

: 6) 5.4.5 Moist v ertical diusion There is an option with explicit moisture of including the eects of moisture on v ertical diusion. T aking in to accoun t moist-adiabatic processes in cloudy air (Durran and Klemp 1982), (5.4.4.5) is modied to R i = (1 + )

"

g S @ @z g

2

 1+  Sc p

T # (5 : 4

: 5

: 1) where  = L

2 v

q

vs c p

R

v

T

2 (5 : 4

: 5

: 2) and

= L

v

q

vs R d

T ; (5 : 4

: 5

: 3) and this modied v alue is used in (5.4.4.2) where the cloud amoun t exceeds 0.01 g kg 1

.

865.5 A tmospheric radiation param eterization The atmospheric radiation option in the model pro vides a longw a

v

e (infra-red) and short w

a

v

e (visible) sc heme that in teract with the atmosphere, cloud and precipitation elds, and with the surface (Dudhia 1989). 5.5.1 Longw av e radiativ e sc heme Longw a

v

e absorption b y w ater v apor, the primary clear-air absorber, is strongly spectral in c haracter, and the method emplo y

ed is the commonly used broadband emissivit y method (see Stephens 1984). This in v

olv es using a precalculated emissivit y function, 

, whic h represen ts the frequency-in tegrated absorption spectrum of w ater v apor, w eigh ted b y a suitable en v

elope function. Rodgers (1967) giv es an up w

ard and do wn w

ard emissivit y as a function of w ater v apor path, u

, with a temperature correction term, where u

includes a pressure correction factor proportional to p

0

: 86 . The form of the tted function is

 (

u

) = i =4 X i =0 (

a

i + T b i

)

x

i

; (5 : 5

: 1

: 1) where x = ln u and T is a u

-w eigh ted T - 250K. F or u less than 10 g m 2

, the form is

 (

u

) = i =4 X i =1 (

c

i + T d i

)

y

i

; (5 : 5

: 1

: 2) where y = u

1

=

2 and a

i

,

b

i

,

c

i

and d

i

are constan ts. In the tropics, e

-t

ype absorption is an importan t additional componen t of the longw a

v

e absorption spectrum and is included with a similar fourth-order polynomial in ln ( ue ) to (5.5.1.1) from Stephens and Webster (1979), where e

is the partial pressure due to w ater v apor. Giv en the emissivit y functions from (5.5.1.1-2) ( 

u

for up w

ard 
ux and 

d

for do wn w

ard 
ux), the up w

ard and do wn w

ard uxes at an y model lev el are giv en b y F u = Z 1 0

B

(

T

)

d u

; (5 : 5

: 1

: 3

a

) F d = Z 1 0

B

(

T

)

d d

; (5 : 5

: 1

: 4

a

) In (5.4.1.3a) the in tegration is performed do wn w

ards through the model la y

ers. The quan tit y d is calculated for eac h la y

er using the temperature ( T

) of the la y

er and the 87frequency-in tegrated Planc k function B = 

SB T

4

, where 

SB is the Stefan-Boltzmann constan t. When the surface is encoun tered, the ground emission F bot is m ultiplied b y 1 

and added to the in tegration. In (5.5.1.4a), the in tegration is performed up w

ards; the do wn w

ard longw a

v

e 
ux at the model top, F top , is assumed to result only from CO 2 emission in the stratosphere. Th us (5.5.1.3a-4a) can be expressed as F u (

z

) = Z z

0 =

z

sf c
z 0 =

z B

(

T

)

d u dz 0

dz 0 + F bot [1 

u

(

z; z sf c )]

; (5 : 5

: 1

: 3

b

) and F d (

z

) = Z z

0 =

z

top
z 0 =

z B

(

T

)

d d dz 0

dz 0 + F top [(1 

d

(

z; z top )]

; (5 : 5

: 1

: 4

b

) where 

(

z; z 1

) = Z z

1
z d dz

0

dz 0

: (5 : 5

: 1

: 5) It can be seen from the form ulas that if the emissivit y reac hes 1 during the in tegration, the remaining atmosphere mak es no con tribution to the 
ux. This is consisten t with the idea that an emissivit y of 1 corresponds to a \blac k" la y

er with respect to longw a

v

e radiation. F ollo wing Stephens (1978), the cloud w ater is assumed to ha v

e a constan t absorption coecien t whic h is sligh tly dieren t for up w

ard and do wn w

ard radiation. The absorption coecien ts are cu = 0 : 130 m 2

g 1

and cd = 0 : 158 m 2

g 1

. To com bine these with w ater v apor absorption, the transmissivities are m ultiplied since clouds are assumed to be \grey bodies." The net emissivit y is then



tot = 1 T v T c ; (5 : 5

: 1

: 6) with T v = 1 

vapor and (5 : 5

: 1

: 7) T c = exp( c

u

c

)

; (5 : 5

: 1

: 8) where u

c

is the cloud w ater path (liquid mass per unit area). Ice cloud is assumed to be composed of hexagonal plate-like crystals with the diametermass relation giv en in section (5.3.1.1). If the assumption is made that the crystals do not re
ect longw a

v

e radiation and are sucien tly thic k to be \blac k", it is possible to estimate 88an absorption coecien t as an in tegrated cross-sectional area. Allo wing for the random orien tation of these crystals and a hemispheric in tegration factor of 1.66, the absorption coecien t tak es a v alue of i

= 0 : 0735 m 2

g 1

, or about half that of cloud w ater. Since this v alue agrees with some observ ations, it w as applied in the model. F or rain and sno w, the size distribution is necessary since the cross section is not proportional to the mass of a particle. The size spectrum c hanges with precipitation in tensit y so the absorption coecien t v aries with precipitation amoun t. The eectiv e absorption coecien t is giv en b y

p

= 1

: 66 2000 

N0  3 r 

1

=

4

m 2

g 1

; (5 : 5

: 1

: 9) where 

r

is the particle densit y

. F or the constan ts used in the explicit moisture sc heme described earlier, the absorption coecien ts tak e v alues of 2 : 3410 3

m 2

g 1

for sno w and 0 : 33010 3

m 2

g 1

for rain. The eectiv e w ater path for a la y

er of  z

meters thic kness is giv en b y u

p = ( q r

)

3

=

4

 z1000gm 2

; (5 : 5

: 1

: 10) so that the transmissivit y is giv en b y T p = exp( p

u

p

)

: (5 : 5

: 1

: 11) This transmissivit y is m ultiplied with the others in (5.5.1.6) to giv e



tot . This is kno wn as an o v

erlap appro ximation. Rain and sno w ha v

e less eect on the longw a

v

e 
ux b y 2 to 3 orders of magnitude, but still are not insignican t. Carbon dio xide is less easily treated since it cannot be assumed \grey". That is, its absorption is concen trated in a band of infrared w a

v

elengths. To include its eect, an o v

erlap method is used as discussed b y Stephens (1984). In eect, the spectrum is divided in to a carbon-dio xide band and a non-carbon-dio xide region. The former requires o v

erlapping of the carbon dio xide transmissivit y function while the latter does not. The relativ e w eigh ts of these t w

o regions is sligh tly temperature dependen t, but they add to giv e the total absorption. A pressure correction factor proportional to p

1

: 75 is applied to the carbon dio xide path calculation. 89Ha ving obtained the 
ux proles, F u (

z

) and F d (

z

), the radiativ e heating rate is calculated from _ Q R = c

p

@T @ t = 1  @ @z (

F d F u ) = g @ @p (

F d F u )

: (5 : 5

: 1

: 12) In the model, the v alues of F are dened on the model full sigma-lev els. This mak es the v arious in tegrals and deriv ativ es easier to represen t n umerically . 5.5.2 Short w

av e Radiativ e Sc heme The do wn w

ard componen t of short w

a

v

e 
ux is ev aluated taking in to accoun t 1) the eects of solar zenith angle, whic h in
uences the do wn w

ard componen t and the path length; 2) clouds, whic h ha v

e an albedo and absorption; 3) and clear air, where there is scattering and w ater-v apor absorption. Th us, S d

(

z

) = S 0 Z top
z (

dS cs + dS ca + dS s + dS a

)

; (5 : 5

: 2

: 13) where  is the cosine of the zenith angle and S

0

is the solar constan t. As with the longw a

v

e sc heme, cloud fraction in a grid bo x is either 0 or 1 because of the assumed stratiform nature of the clouds. The cloud bac k-scattering (or albedo) and absorption are bilinearly in terpolated from tabulated functions of  and ln( w= ) (where w is the v ertically in tegrated cloud w ater path) deriv ed from Stephens' (1978) theoretical results. The total eect of a cloud or m ultiple la y

ers of cloud abo v

e a heigh t

z is found from the abo v

e function as a percen tage of the do wn w

ard solar 
ux absorbed or re
ected. Then at a heigh t z  z

, a new total percen tage is calculated from the table allo wing the eect of the la y

er  z to be estimated. Ho w

ev er, this percen tage is only applied to S 0  S

(

clear air ); that is, the clear-air eect abo v

e

z

is remo v

ed. Clear-air w ater v apor absorption is calculated as a function of w ater v apor path allo wing for solar zenith angle. The absorption function is from Lacis and Hansen (1974). The method is a similar in tegration-dierence sc heme to that described abo v

e for cloud. Clear-air scattering is tak en to be uniform and proportional to the atmosphere's mass path length, again allo wing for the zenith angle, with a constan t giving 20 percen t scattering in one atmosphere. The heating rate is then giv en b y R T = R

T

(

longwave ) + 1 c

p

S

abs ; (5 : 5

: 2

: 14) 90where S

abs is dened from the absorption part of the S

d

in

tegral giv en in (5.5.2.13), since only cloud and clear-air absorption con tribute to solar heating. The solar and infrared 
uxes at the surface, calculated from the atmospheric radiativ e sc hemes, are use in the energy budget of the land surface.

91Appendix 1. Glossary of Sym bols

a Fraction of con v

ectiv e cloud co v

er; also constan t used in cloud microph ysics ABE Av ailable buo y

an t energy A T The forcing terms of the thermodynam ic equation that v ary on the time-scale of the Rossb y-w a

v

es A u

; A d

; A tot Cloud Work Function for updraft, do wndraft, and all of model cloud A v The forcing terms of the v-momen tum equation that vary on the timescale of the Rossb y-w a

v

es A v The forcing terms of the u-momen tum equation that vary on the timescale of the Rossb y-w a

v

es A 0 P

arameter for heterogeneous freezing (K 1

) A An tidiusiv e 
ux b Bac kscattering coecien t; also fraction of total w ater v apor con v

ergence used to moisten grid column (section 5.3.2.1); also constan

t (0.8) used in cloud microph ysics computation B Planc k function B u Acceleration due to buo y

ancy B 0 P

arameter for heterogeneous freezing (m 3

s 1

)

c 0 Rainfall con v

ersion parameter (section 5.3.2.2) 93c m Constan t used in computation of D

m

c i Coecien ts used in calculation of cloud eect on do wn w

ard longw a

v

e radiation (T able 5.2) c p Specic heat at constan t pressure for dry air

c pm Specic heat at constan t pressure for moist air

c  Net condensation rate a v

eraged o v

er grid v olume

c  c Net condensation rate in cum ulus cloud (section 5.3.2.1) C Constan t (2. m s 1

K 1

=

2

) used in computing con v

ectiv e v elocit y C g Thermal capacit y of slab per unit area (J m 2

K 1

) C s Heat capacit y per unit v olume (J m 3

K 1

) C  Surface exc hange coecien t for heat C u Surface exc hange coecien t for momen tum ; also total condensate in updraft (section 5.3.2.2) C D Surface drag coecien t C 0 D Componen t of surface-drag coecien t C uN Value of surface mom en tum exc hange coecien t under neutral stabilit y conditions 94C N Value of surface heat exc hange coecien t under neutral stabilit y conditions D Mass div ergence (h ydrostatic split-explicit sc heme); also horizon tal deformation (section 5.1) D f Diusivit y of w ater v apor in air D u

; D d

; D tot Updraft, do wndraft, and total cloud kinetic energy dissipation Distance bet

w

een an observ ation and a giv en grid poin t (section 4) D i Diameter of ice crystal (m) D Diusion and PBL tendencies for v ariable D m Modied distance bet w

een an observ ation and a giv en grid poin t (section 4) e Horizon tal Coriolis parameter (s 1

)

e s

; e si

; e sw Saturation v apor pressure, o v

er ice, o v

er w ater (cb) E Eciency of collection of cloud b y precipitation; also v ertical 
ux of w ater v apor E s Flux of w ater v apor from surface in to atmosphere f Coriolis parameter f1 ; f2 Ven tilation coecien ts for rain or sno w

95F Larger-scale forcing (section 5.3.2.2 and 5.3.2.3); also function of distance from lateral boundary (section 2.6.2) FH; F L Flux from high-order and lo w-order adv ectiv e sc heme F bot ; F top Longw a

v

e radiativ e 
ux at bottom, top of model atmosphere (W m 2

) F d ; F u Do wn w

ard, up w

ard longw a

v

e radiativ e 
ux (W m 2

) F H T

erm represen ting con tribution of horizon tal diusion of a v ariable to the temporal rate of c hange of F V T

erm represen ting con tribution of v ertical diusion of a v ariable to the temporal rate of c hange of F s Flux of dry static energy (section 5.3.2.2); also Surface 
ux of heat, moisture or mom en tum F q Flux of w ater v apor (section 5.3.2.2) F l Flux of suspended cloud liquid w ater (section 5.3.2.2) F r Deceleration F 1 ; F 2 Amplitude factors used in computing lateral boundary conditions (section 2.6.2) g Acceleration of gra vit y (9.8 m s 2

) h Moist static energy; also heigh t of planetary boundary la y

er (m) h o Local hour angle of sun

96~ h; h c

; h u

; h d

;

~ h

 Moist static energy in en vironmen t, cloud, updraft, do wndraft, and saturation v alue in en vironmen t H Vertical 
ux of sensible heat (W m 2

) H m Heat 
ux in to substrate (W m 2

) H s Sensible heat 
ux from surface in to atmosphere (W m 2

) I Function of static stabilit y and surface friction v elocit y; also horizon tal grid-index in y-direction IMAX Maxim um v alue of grid-index in y-direction Is Net longw a

v

e iradiance at surface ( wm 2

) I " Outgoing longw a

v

e radiation from surface (W m 2

) I # Do wn w

ard longw a

v

e radiation absorbed at surface (W m 2

) under clear skies I #

0 Do wn w

ard longw a

v

e radiation absorbed at surface (W m 2

) in presence of clouds I1 Normalized condensate in updraft (section 5.3.2.2) I2 Normalized ev aporate in updraft (section 5.3.2.2) J Horizon tal grid index in x

-direction JMAX Maxim um v alue of grid index in x

-direction

97k Dimensionless x

-w a

v

en um ber for upper radiativ e sc heme; also v on Karman constan t (0.4) ^ k Dimensionless eectiv e x

-w a

v

en um ber for upper radiativ e sc heme k 1 Constan t used in form ula for computing autocon v

ersion of cloud drops to rain drops K Total horizon tal w a

v

enm u

m

ber (m 1

) also Kernels K a Bac kground molecular diusivit y (2 : 410 5

m 2

s 1

); also thermal conductivit y of air ( J m 1

s 1

K 1

) K H Horizon tal eddy diusivit y (m 2

s 1

) K 0 H Coecien t used in fourth-order diusion (s 1

) K HO Bac kground v alue of horizon tal eddy diusivit y (m 2

s 1

) K m Coecien t of heat transfer from ground in to substrate (s 1

) KMAX Maxim um v alue of index in v ertical direction K z Coecien t of v ertical diusivit y (m 2

s 1

) K z

0 Bac kground v alue of coecien t of v ertical diusivit y (m 2

s 1

) KE u

;KE d

;KE tot Kinetic energy for updraft, do wndraft, and all of model cloud L Hydrostatic term due to liquid w ater loading; also Monin-Obukho v length L m Laten t heat of fusion (0.3510 6

J kg 1

) 98L s Laten t heat of sublimation (2.8510 6

J kg 1

) L v Laten t heat of condensation (2.510 6

J kg 1

) l Dimensionless y

-w a

v

en um ber for upper radiativ e sc heme; also v ertical mixing length ^ l Dimensionless eectiv e

y

-w a

v

en um ber for upper radiativ e sc heme Mi Mass of ice crystal (kg) Mmax Maxim um mass of ice crystal (kg) M0 Initial mass of ice crystal (kg)

m Mass 
ux (updraft and do wndraft) in con v

ectiv e parameterization cloud (5.3.2.2); also map scale factor
 m Mixing coecien t used in free-con v

ectiv e regime of high-resolution PBL model m b Cloud base mass 
ux (section 5.3.2.2)

m 0 Do wndraft base mass 
ux (section 5.3.2.2)

m u Updraft mass 
ux in con v

ectiv e parameterization cloud (5.3.2.2)

m d Do wndraft mass 
ux in con v

ectiv e parameterization cloud (5.3.2.2) M Surface moisture a v

ailabilit y
99M t Vertical in tegral of horizon tal con v

ergence of w ater v apor

n F

raction of cloud

n 0 Cloud microph ysics parameter

n c Num ber concen tration of ice crystals (kg 1

) N Brun t-Vais al a frequency (s 1

) Nc Num ber concen tration of cloud droplets per unit v olume (10 10 m 3

) N h Nondimensional function for v ertical prole of con v

ectiv e heating N m Nondimensional function for v ertical prole of con v

ectiv e moistening N0 Cloud microph ysics parameter (810 6

m 4

for rain 210 7

m 4

for sno w) p Pressure (cb)
p b Pressure (cb) at con v

ectiv e cloud base p s Surface pressure (cb)
p t Pressure (cb) at top of model
p u Pressure (cb) at top of con v

ectiv e cloud
p LCL Pressure (cb) at lifting condensation lev el
100p  p

s p

t

(cb)
p  d Dot-poin t

p



(cb)
p 0 Reference-state pressure p 0 P

erturbation pressure (P a) Pressure v

alue represen ting the free atmosphere, where terrain in
uences are small (in FDD A) ^ p F

ourier transform of p

0

for upper radiativ e boundary condition PCON Condensation of w ater v apor or ev aporation of cloud drops (kg kg 1 s 1

) P RA Accretion of cloud drops b y rain drops (kg kg 1

s 1

) P RC Autocon v

ersion of cloud drops to rain drops (kg kg 1

s 1

) P RE Ev aporation of rain drops (kg kg 1

s 1

) PCI Heterogeneous freezing of cloud w ater (kg kg s 1

) P ID Deposition of v apor on to ice crystals (kg kg s 1

) P I I Initiation of ice crystals (kg kg s 1

) P MF Melting/freezing of cloud and precipitation due to adv ection (kg kg s 1

) 101P RM Melting of falling precipitation (kg kg s 1

) PSM Melting of falling sno w (kg kg s 1

)

~ q; q u

; q d

;

~ q

 Water v apor mixing ratio in en vironmen t, updraft, do wndraft, and saturation v alue in en vironmen t q c Mixing ratio of cloud w ater; also w ater v apor mixing ratio in cloud (section 5.3.2.2) q c

0 Critical v alue of mixing ratio of cloud w ater q r Mixing ratio of rain w ater q l Suspended liquid w ater v apor mixing ratio inside updraft q v Mixing ratio of w ater v apor q vc Mixing ratio of w ater v apor in cum ulus cloud
q vs Saturation mixing ratio of w ater v apor Q Diabatic heating rate per unit mass (J kg 1

s 1

) Q s Net short w a

v

e irradiance at the surface (W m 2

) R Rainfall (con v

ectiv e-scale sink of cloud w ater, 5.3.2.2); also ideal gas constan t for dry air (287 J kg 1

K 1

) RH Relativ e h umidit y R i Ric hardson n um ber 102R n Net radiation R iB Bulk Ric hardson n um ber R ic Critical v

alue of bulk Ric hardson n um ber; also critical v

alue of Ric hardson n um ber R v Gas constan t for w ater v apor (461.5 J kg 1

K 1

) R T Radiativ e heating rate (K s 1

)

r Radius of con v

ectiv e parameterization cloud (sections 5.3.2.2) S Supersaturation; also source or sink term (section 5.3.2.2); also square of the v ertical wind shear S c Sc hmidt n um ber S o Solar constan t (1395.6 W m 2

) S u Source or sink term in updraft (section 5.3.2.2) S d Do wn w

ard solar 
ux (W m 2

); also source or sink term in do wndraft (section 5.3.2.2) S i Supersaturation o v

er ice

s Dry static energy t Time (s)

103T Temperature (K) T c Longw a

v

e transmissivit y due to cloud T d Dewpoin t temperature (K) T g Temperature (K) of ground T p Longw a

v

e transmissivit y due to precipitation T v Virtual temperature (K); also longw a

v

e transmissivit y due to v apor T  Surface friction temperature (K) T 0 Reference-state temperature (K) T 0 P

erturbation temperature (K)

u Componen t of wind v elocit y in east w

ard direction (m s 1

); also w ater v apor path (g m 2

) u  Surface friction v elocit y (m s 1

)

u c

; u p Liquid w ater path for cloud, precipitation (g m 2

)

v Componen t of wind v elocit y in north w

ard direction (m s 1

)

v t Mass w eigh ted mean terminal v elocit y of rain drops (m s 1

) V F

all speed of a precipitation particle (m s 1

); also modied horizon tal wind v elocit y in PBL 104V Horizon tal wind v ector V a Horizon tal windspeed at lo w

est model la y

er V c Con v

ectiv e PBL v elocit y(m s 1

) V qf Div ergence of v ertical eddy 
ux of w ater v apor due to con v

ectiv e clouds w Vertical v elocit y (m s 1

); also w eigh t function for reducing mixing near top of mixed la y

er w n Weigh t function for blending model tendencies and large-scale tendencies near lateral boundaries (section 2.6.1) w p Precipitable w ater (cm)

w u Vertical v elocit y in updraft
^ w F

ourier transform of w

x Horizon tal grid coordinate increasing generally east w

ard X Horizon tal coordinate on earth surface increasing generally east w

ard X c Multiple-re
ection factor in cloudy air X d Distance v ector X R Multiple-re
ection factor in clear air

105y Horizon tal grid coordinate increasing generally north w

ard Y Horizon tal coordinate on earth surface increasing generally north w

ard

z Heigh t abo v

e surface (m)

z a Heigh t of lo w

est la y

er in model (m)

z b Heigh t of updraft originating lev el(section 5.3.2.2) (m)

z 0 Heigh t of do wndraft originating lev el(section 5.3.2.2); also surface roughness length(m) z oc Bac kground v alue of surface roughness length o v

er w ater (10 4

m)

z l Depth of molecular la y

er

z LCL Heigh t of lifting condensation lev el (m)

z T Heigh t of updraft top (section 5.3.2.2) Coecien t arra y for upper radiativ e boundary condition (m s 1 P a 1

); also an y thermodynamic v ariable (section 5.3.2.2) ~ An y thermodynam ic v ariable in en vironmen t

u An y thermodynam ic v ariable in updraft

d An y thermodynam ic v ariable in do wndraft

c ; p Longw a

v

e absorption coecien ts for cloud, precipitation (m 2

g 1

) 106P arameter in sound-w a

v

e temporal dierencing; also precipitation eciency parameter in section 5.3.2.2 Gamma function

d Dry adiabatic lapse rate (K m 1

)
dp Dewpoin t adiabatic lapse rate (K m 1

) Ratio of heat capacities ( c

p

=c v

) for dry air  Solar declination M Supersaturation or undersaturation  p Vertical grid size (P a)  s Horizon tal grid length (m)  t Time step (s)  t

0 Short time step for rain fall term (s)  x Horizon tal grid length (m)  z Thic kness of v ertical la y

er (m)   Thic kness of model  lev els

107 

c Critical v alue of con v

ectiv e cloud depth   Short time step for sound w a

v

es (s) r 2  Horizon tal Laplacian on 

-surfaces r 4  F

ourth order diusion operator on 

-surfaces

 P

arameter relating updraft and do wndraft mass 
ux (section 5.3.2.2); also small v alue; also en trainmen t coecien t used in high resolution PBL-model (0.2)  a A

tmospheric emissivit y

 g Emissivit y of ground

 u

; 

d A

tmospheric longw a

v

e emissivit y  d Normalized mass 
ux for do wndraft (section 5.3.2.2)
 u Normalized mass 
ux for updraft (section 5.3.2.2)  P

oten tial temperature (K); also angle bet w

een y

-axis and north for full Coriolis force  a P

oten tial temperature (K) at lo w

est la y

er in model  g P

oten tial temperature (K) of ground surface  e Equiv alen t poten tial temperature (K)

108 es Saturation equiv alen t poten tial temperature (K)  v Virtual poten tial temperature (K)  Longitude; also cloud t ype (section 5.3.2.2); also thermal conductivit y (J m 1

s 1

K 1

); also parameter in raindrop distribution (m 1

)  Dynamic viscosit y of air (kg m 1

s 1

); also solar zenith angle ; also total net fractional en trainmen t rate (section 5.3.2.2); also constan t in smoother (section 3.3)  u T

otal net fractional en trainmen t rate for updraft (section 5.3.2.2)
 ue Gross fractional en trainmen t rate for updraft (section 5.3.2.2)
 ud Gross fractional detrainmen t rate for updraft (section 5.3.2.2)

 Coecien t for Asselin time lter; also for spatial smoother

 Exner function  Densit y of air (kg m 3

)
 r P

article densit y (kg m 3

)
 u Densit y in updraft  w Densit y of w ater (kg m 3

)
 0 Reference-state densit y (kg m 3

)

109 0 P

erturbation densit y (kg m 3

)

 Nondimensional v ertical coordinate of model

 0 Dumm y v ariable of in tegration
_  Vertical v elocit y in 

-coordinates (s 1

)

_  c Vertical v elocit y of con v

ectiv e cloud in 

-coordinates (s 1

)

 SB Stefan-Boltzmann constan t (5 : 6705110 8

J m 2

K 4

s 1

)

 Half-period of time windo w of in
uence of an observ ation (section 4); also short-w a

v

e transmissivit y  0 Short-w a

v

e transmissivit y obtained from lookup table

a Clear air absorption transmissivit y

ac Cloudy air absorption transmissivit y

 s Clear air scattering transmissivit y; also surface stress

 sc Cloudy air scattering transmissivit y  Geopoten tial; also latitude; also scalar v ariable in adv ection equation  s Surface geopoten tial

110 Sym bol denoting lo w-order, monotonic solution to adv ection equation
 Diusivit y of v apor in air (m 2

s 1

); also thermal inertia Solar zenith angle; also function of bulk Ric hardson n um ber

m Nondimensional stabilit y parameter for momen tum

h Nondimensional stabilit y parameter for heat and w ater v apor

! Vertical v elocit y in pressure coordinates (cb s 1

)

! c Vertical v elocit y of con v

ectiv e cloud in pressure coordinates (cb s 1

) Angular v elocit y of earth (7 : 272210 5

s 1

)

111

# Appendix 2. Look-up table for transmissivities

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 1.0 0.926 0.868 0.855 0.846 0.838 0.832 0.827 0.822 0.818 0.814 0.811 1.2 0.915 0.854 0.840 0.831 0.823 0.817 0.811 0.806 0.802 0.798 0.794 1.4 0.903 0.841 0.826 0.816 0.808 0.802 0.796 0.791 0.787 0.782 0.779 1.6 0.892 0.828 0.813 0.803 0.795 0.788 0.782 0.777 0.772 0.768 0.764 1.8 0.881 0.815 0.800 0.790 0.782 0.775 0.769 0.763 0.758 0.754 0.750 2.0 0.870 0.803 0.788 0.777 0.769 0.762 0.756 0.750 0.745 0.741 0.737 2.2 0.860 0.792 0.776 0.765 0.757 0.750 0.743 0.738 0.733 0.728 0.724 2.4 0.850 0.781 0.765 0.754 0.745 0.738 0.731 0.726 0.7210 0.716 0.712 2.6 0.839 0.770 0.753 0.742 0.733 0.726 0.720 0.714 0.709 0.704 0.700 2.8 0.830 0.759 0.743 0.731 0.722 0.715 0.709 0.703 0.698 0.693 0.689 3.0 0.820 0.748 0.732 0.721 0.712 0.704 0.698 0.692 0.687 0.682 0.678 3.2 0.810 0.738 0.722 0.710 0.701 0.694 0.687 0.681 0.676 0.671 0.667 3.4 0.801 0.728 0.712 0.700 0.691 0.683 0.677 0.671 0.666 0.661 0.656 3.6 0.791 0.719 0.702 0.690 0.681 0.674 0.667 0.661 0.656 0.651 0.646 3.8 0.782 0.709 0.692 0.681 0.671 0.664 0.657 0.651 0.646 0.641 0.636 4.0 0.773 0.700 0.683 0.671 0.662 0.654 0.648 0.642 0.636 0.631 0.627 4.2 0.764 0.691 0.674 0.662 0.653 0.645 0.638 0.632 0.627 0.622 0.618 4.4 0.756 0.682 0.665 0.653 0.644 0.636 0.629 0.623 0.618 0.613 0.608 4.6 0.747 0.673 0.656 0.644 0.635 0.627 0.621 0.615 0.609 0.604 0.600 4.8 0.738 0.665 0.647 0.636 0.626 0.619 0.612 0.606 0.600 0.596 0.591 5.0 0.730 0.656 0.639 0.627 0.618 0.610 0.603 0.597 0.592 0.587 0.582 5.2 0.722 0.648 0.631 0.619 0.610 0.602 0.595 0.589 0.584 0.5790 0.574 5.4 0.714 0.640 0.623 0.611 0.602 0.594 0.587 0.581 0.576 0.571 0.566 5.6 0.706 0.632 0.615 0.603 0.594 0.586 0.579 0.573 0.568 0.563 0.558 5.8 0.698 0.624 0.607 0.595 0.586 0.578 0.571 0.565 0.560 0.555 0.550 6.0 0.690 0.616 0.599 0.588 0.578 0.571 0.564 0.558 0.552 0.547 0.543 6.2 0.683 0.609 0.592 0.580 0.571 0.563 0.556 0.550 0.545 0.540 0.535 6.4 0.675 0.602 0.585 0.573 0.564 0.556 0.549 0.543 0.538 0.533 0.528 6.6 0.668 0.594 0.577 0.566 0.556 0.549 0.542 0.536 0.531 0.526 0.521 6.8 0.661 0.587 0.570 0.559 0.549 0.542 0.535 0.529 0.524 0.519 0.5147 7.0 0.653 0.580 0.563 0.552 0.542 0.535 0.528 0.522 0.517 0.512 0.507 7.2 0.646 0.573 0.556 0.545 0.536 0.528 0.521 0.515 0.510 0.505 0.501 7.4 0.639 0.567 0.550 0.538 0.529 0.521 0.515 0.509 0.503 0.499 0.494 7.6 0.633 0.560 0.543 0.532 0.522 0.515 0.508 0.502 0.497 0.492 0.488 7.8 0.626 0.553 0.537 0.525 0.516 0.508 0.502 0.496 0.491 0.486 0.481 8.0 0.619 0.547 0.530 0.519 0.510 0.502 0.496 0.490 0.484 0.480 0.475 8.2 0.613 0.541 0.524 0.512 0.503 0.496 0.489 0.484 0.478 0.473 0.469 8.4 0.606 0.534 0.518 0.506 0.497 0.490 0.483 0.477 0.472 0.467 0.463 8.6 0.600 0.528 0.512 0.500 0.491 0.484 0.477 0.472 0.466 0.462 0.457 8.8 0.594 0.522 0.506 0.494 0.486 0.478 0.472 0.466 0.461 0.456 0.451 9.0 0.587 0.516 0.500 0.489 0.480 0.472 0.466 0.460 0.455 0.450 0.446 9.2 0.581 0.511 0.494 0.483 0.474 0.467 0.460 0.454 0.449 0.445 0.440 9.4 0.575 0.505 0.488 0.477 0.468 0.461 0.455 0.449 0.444 0.439 0.435 9.6 0.569 0.499 0.483 0.472 0.463 0.456 0.449 0.444 0.438 0.434 0.429 9.8 0.563 0.494 0.477 0.466 0.458 0.450 0.444 0.438 0.433 0.428 0.424 10.0 0.558 0.488 0.472 0.461 0.452 0.445 0.439 0.433 0.428 0.423 0.419 

# precipitable water (cm)

Path length

112Appendix 3. Map Pro jections Map pro jections are constructed b y pro jecting the surface of the earth on to a righ t circular cone, cutting the cone, and 
attening it in to a plane surface. Three pro jections are a v

ailable for the MM4 system { P olar stereographic, Lam bert conformal, and Mercator. P olar stereographic is preferred for high-latitude studies, Lambert conformal for middlelatitude studies, and Mercator for lo w-latitude studies. This appendix summa rizes the map scale factors for eac h pro jection and giv es the equations for con v

erting from latitude and longitude to the x and y positions on the model grid. Although the grid size  x =  y =  s

is constan t on the model's grid, the actual distance represen ted b y  s

on the spherical earth v aries with location on the grid because the earth is curv ed. The map scale factor m is dened as the ratio of the distance on the grid to the corresponding distance on the earth's surface
m = distance on grid actual distance on earth A: 1 a. Lam bert Conform a

l Conformal means that the scale is equal in all directions about a poin t, so that shapes of geographic features on the earth are preserv ed. The Lam bert conformal grid is true at latitudes 30 

and 60 

N so that m = 1 :

at these latitudes. In general,
m = sin 1 sin 

tan= 2 tan 1

=

2



0

: 716 ; A: 2 where 1

= 30 

and is the colatitude ( = 90  

). It is sometimes necessary to compute the position ( x; y ) on the grid giv en the latitude and longitude of a poin t, or vice v ersa. The follo wing relations pertain to an X Y grid with cen ter X = 0 ; Y = 0 at latitude 

0

and longitude 

0

. Note that the relationship bet w

een ( x; y ) and ( X; Y ) is
x = X + JMAX 1 2

 s; A: 3 y = Y + IMAX 1 2

 s; A: 4  = any longitude 113 0

= longitude of Y axis  = any latitude  0

= latitude along  0

at which Y = 0 = 90  
n = : 716
1 = 30 
0 = 90  

0 a = 6370 km
r = a n

sin 1



tan = 2 tan 1

=

2



n

; A: 5 C 2

= a n

sin 1



tan 0

=

2 tan 1

=

2



n

; A: 6 C 1

= 

0 90 

=n; A: 7  0

= n

(



+ C

1

)

; A: 8 X = rcos 0

; A: 9 Y = rsin 0

+ C

2

: A: 10 The in v

erse problem to calculate latitude and longitude is done as follo ws:  0

= arctan 

Y C

2 X

 ; A: 11  = 

0 n C

1

; A: 12
r = X cos 0

or Y C

2 sin 0 A: 13 = 2 arctan "

tan 1

=

2

 nr asin 1



1

=n #

; A: 14  = 90  : A: 15 b. P olar Stereographic 114For the polar stereographic pro jection, true at latitude 

1 = 60 

N , the map scale factor is m = 1 + sin 1 1 + sin A: 16 The relationships bet w

een latitude and longitude and X and Y on the polar stereographic grid are calculated as before on the Lam bert conformal grid, but no w n = 1.

r = amcos; A: 17 C 2

= asin 1



tan 0

=

2 tan 1

=

2



; A: 18 C 1

= 

0 90 

; A: 19  0

= 

+ C

1

; A: 20 X = rcos 0

; A: 21 Y = rsin 0

+ C

2

: A: 22 and for the in v

erse problem 

0

= arctan 

Y C

2 X

 ; A: 23  = 

0 C

1

; A: 24
r = X cos 0

or Y C

2 sin 0 A: 25 = 2 arctan 

tan 1

=

2

 r asin 1

 ; A: 26  = 90  : A: 27 Note that the signs of Y C

2 and X in (A.23) m ust be considered to obtain the correct quadran t for 

0 . c. Mercator F or the Mercator grid, 

0

(

Y = 0) corresponds to the equator and the relationships bet w

een X and Y and  and  are relativ ely simple X = ( acos 1

)(

 

0

)

; A: 28 115Y = ( acos 1

)

ln 

1 + sin cos 

; A: 29 a Y = ( acos 1

)

ln [ tan (45 

+ = 2)] : A: 29 b Note that (  

0

) in (A.28) m ust be expressed in radians. The latitude 

1

at whic h the Mercator pro jection is true is often tak en to be 30 

. The rev erse problem, to obtain X and Y from  and 

, is also simple  = 

0

+ X acos 1

: A: 30 To solv e for 

, use (A.29b)  = 90 

+ 2 arctan 

exp  Y acos 1



: A: 31

116

# Appendix 4. Land Use Categories Description of land-use categories and physical parameters for summer (15 April-15 October) and winter (15 October-15 April).

Landuse Integer Identification Landuse Description Albedo(%) Moisture Avail. (%) Emissivity (% at 9μm)Roughness Length (cm) Thermal Inertia (cal cm-2

# k-1

# s-1/2

# ) Sum Win Sum Win Sum Win Sum Win Sum Win 1 Urban land 18 18 5 10 88 88 50 50 0.03 0.03 2 Agriculture 17 23 30 60 92 92 15 5 0.04 0.04 3 Range-grassland 19 23 15 30 92 92 12 10 0.03 0.04 4 Deciduous forest 16 17 30 60 93 93 50 50 0.04 0.05 5 Coniferous forest 12 12 30 60 95 95 50 50 0.04 0.05 6 Mixed forest and wet land 14 14 35 70 95 95 40 40 0.05 0.06 7 Water 8 8 100 100 98 98 .01 .01 0.06 0.06 8 Marsh or wet land 14 14 50 75 95 95 20 20 0.06 0.06 9 Desert 25 25 2 5 85 85 10 10 0.02 0.02 10 Tundra 15 70 50 90 92 92 10 10 0.05 0.05 11 Permanent ice 55 70 95 95 95 95 5 5 0.05 0.05 12 Tropical or sub tropical forest 12 12 50 50 95 95 50 50 0.05 0.05 13 Savannah 20 20 15 15 92 92 15 15 0.03 0.03

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