Celal S. Konor Release 1.1 (identical to 1.0) 3/21/08. 1-Hybrid isentropic-sigma vertical coordinate and governing equations in the free atmosphere

Transcription

1 Celal S. Konor Release. (identical to.0) 3/2/08 -Hybrid isentropic-siga vertical coordinate governing equations in the free atosphere This section describes the equations in the free atosphere of the odel. We first discuss the generalized vertical coordinate describe the - hybrid vertical coordinate selected for the odel. Then we introduce the vertical ass flux equation unique for such a generalized vertical coordinate briefly discuss the upper lower boundary conditions. We next describe the vertical grid vertical discretization in the free atosphere. The equations governing the dry dynaics closely follow Konor Arakawa (997).. Vertical coordinate The vertical coordinate is defined by F(, ) () Gp, ( ). (2) Since is the vertical coordinate, it is required that F(,) be a onotonic function of height. Here we select F as a onotonically increasing function of height, therefore, the ost straightforward choice for G is the sae. Now we ake the following choice F(, ) f ( )+ g( ) (3) where g( ) g o ( e ), (4) g o e T (5) = where T = Gp ( T, ) at the top of the atosphere Equations (4) (5) yield g T p T = constant g( )= 0 at the botto of the free atosphere (or at the PL-top). Note that the function g( ) soothly exponentially approaches to with height. The rate of approach is controlled by the constant, which is currently 0. Requiring F > 0 guaranties the onotonousness of with height, which requires df d + dg d + g > 0 fro (3). Since g > 0 d g d > 0 between T, we can satisfy the requireent by replacing d d with their unreachably sall values in d d in, respectively. Then we write

2 Celal S. Konor Release. (identical to.0) 3/2/08 d f d + d g in d + g g = 0. (6) in In (6), d g d = g oe = ( g o g ) d f, where f is the unknown. y vertically d integrating (6) with respect to, we obtain f ( ) = in where we used f T expression for the vertical coordinate as g o ( T ) ( g) ( ) in ( g), (8) = 0 g( T ) =. Using (3), (4) (8) in (), we obtain an F(, ) = in + [ in ]g Definition for G( p, ): in g o ( T ) ( g). (9) So far no specific definition is needed for Gp,p ( ). Here we discuss a couple of choices for Gp,p ( ). An obvious choice for G( p, ) is a linearly decreasing function of p, which can be written as p Gp, p T. (0) Since this expression is very siple straightforward, we use it in the developent stage of the odel. With (0), the partial derivatives becoe G = p p p T G p = p p T 2 p T. () However, when (0) is used, dependency of on does not vanish copletely in the iddle upper odel atosphere, which is not desirable in a GCM with a variable PL since it ay cause excessive vertical dissipation of oisture tracers. In the next subsection, we discuss a different expression for G( p, ), which liits the dependency of on near the PL top eliinates it entirely in the iddle upper odel atosphere. A new definition for G( p, ): To define G, we first define 2

3 Celal S. Konor Release. (identical to.0) 3/2/08 p T G 2 G 0 p C p, 0 p T p p C for p p C, (2) p p C G 0 λ λ 0 - p C 0 - p T G where p C is a constant pressure, aybe chosen slightly saller then a possible iniu of, p C = p S - 200b is a reasonable selection, 0 is a stard constant value of. Then G 2 ( p) 0 p for p 0 p C p p T. (3) T Now for siplicity, we define p. (4) p C y using (4) in (2) (3), we respectively write G ( ) 0 p C 0 p T for 0 (5) G 2 (, ) 0 + p C 0 p T 0 p T for p T (6) p C Now we require that G = a G with a + b =. We select a 2 tanh + b G2, (7) { [ ( C )]} it results in b { 2 + tanh [ ( C )]}. With this choice, we satisfy the following G G as 0, (8) p T G G 2 as p T (9) p C p p C G G G 2 G = 2 G + G 2 for = C. (20) 0 λ λ 0 - p C 0 - p T G 3

4 Celal S. Konor Release. (identical to.0) 3/2/08 We can satisfy (7) by Gp, ( ) 2 p p C ncosh p C p p C + p C p p T p C ncosh p C p p C + C, (2) 0 p C 0 p T where we choose C ncosh 2 (22) 0 p T 0 to satisfy G = 0 for p =. Equations (2) (22) yield G p 2 0 p C 0 p T + p C tanh p p C p C p C + 2 p C 0 p T p C tanh p C p p C p C (23) + 2 G 0 p C p C p + tanh p p C 2 p 0 p T ( p C ) 2 p C + 0 p T ncosh p C p p C p C p tanh p C p p C p C 2( 0 p T ) ncosh ( ) (24).2- Generalized vertical ass flux equation Since is the vertical coordinate of the odel, we require 0 = t F(, ). (30) Then using () (2) in (30), we obtain 0 = F t + F. (3) t 4

5 Celal S. Konor Release. (identical to.0) 3/2/08 In (3), ( t) can be obtained fro (2) as t = G p Using (32), equation (3) can be rewritten as 0 = F t + F p t + G G p The therodynaic equation for the syste is t p t + F p. (32) G t. (33) t = v + Q. (34) The pressure tendency surface pressure tendency equations for the syste are p t = ( v )d = + T p (35) t = ( v)d + = T, (36) respectively. y using (34), (35) (36) in (33), the equation that deterines the generalized vertical ass flux can be obtained as F F G = F p v + Q + F G p ( v )d p = T + G p v d + = T p. (37) The vertically discrete version of the generalized vertical ass flux equation (37) will be discussed later in this text..3 Upper Lower boundary conditions The upper boundary (upper ost interface, T ) is an isentropic surface ( T T = constant ). We assue that p T = constant T = 0 {i.e. 5

6 Celal S. Konor Release. (identical to.0) 3/2/08 T = Q = 0}. The lower boundary (lowest interface, ), which coincides with the T PL top, is a siga type, = f( ) = constant. The vertical ass flux ( ) is priarily deterined fro PL top entrainent/detrainent cuulus ass flux fro PL into cuulus clouds..4 Vertical grid in the free atosphere Vertical Structure of the Model in the Free At. T /2 p=p T θ q ~ =0 Q=0 l-/2 l l+/2 p p θ q ~ θ q ~ Q C Q C Free Atosphere L- L-/2 L p θ q ~ Q C L+/2 p θ q ~ Q C L+/2 L+ p θ L q L ~ L Q L R L L+3/2 p θ q ~ Q R PL S M+/2 =0 Fig.. Vertical grid used in the discretization. 6

7 Celal S. Konor Release. 3/2/08.5- Vertically discrete equations in the free atosphere.5.a. Mass continuity equation The vertically discrete version of the ass continuity equation applied to the odel layers within the free atosphere is given by t + v = 0 for =,2,..,L, (38) where ( ) p ( p) p + 2 p 2 ( ) for =,2,...,L, (39) is the vertical ass flux carried at the interfaces of the odel layers. We +2 = 0. The vertical ass flux at the 2 ( ), where L + 2 are interchangeable, is deterined by L +2 assue that, at the top of the atosphere, PL-top, entrainent/detrainent paraeterization. The vertically discrete pressure tendency equations can be obtained by vertically suing (38) with (39) as p +2 t t = ( k v k )( ) k + L k = = k v k k + k = +2 for =,2,...,L 40.a, 40.b where we assued that ( p T t)= 0, where subscript 2 T are interchangeable..5.b. Therodynaic equation The vertically discrete version of the therodynaic equation within the free atosphere applied to the interfaces of the odel layers are given by 7

8 Celal S. Konor Release. 3/2/08 2 = 0 ( 4.a) t +2 + ( v) t L +2 t v L L = L + 2 ( Q ) + 2 for =,2,...,L ( 4.b) = ( Q ) L +2 ( 4.c) Equation (4.a) is a result of choosing the upper boundary placed on an isentropic surface. In (4.b) (4.c),. ( v ) v + + ( ) v + + ( ) + 2 for =,2,...,L, (4.d) ( + ) [ ] ( ) for =,2,...,L, (4.e) where + 2p +2 2 p 2 + ( p + 2 p 2 ) for =,2,...,L, (4.f) L + 2 L + 2 ( ) L + 2 ( ˆ L + 2 L + 2 ) + L + 2 ( L + 2 L )( ) L, (4.g) ˆ L +2 L + 2 if ˆ L L +2 L + 2 if L + 2 < 0 ( ) L + 2 > 0. (4.h) In (4.h), superscript L denotes values fro the upper ost level of the PL, which is also indexed by L + 2. In the equations above, we use + 2 c p p +2 p o. is necessary, p p o c p, if it When we consider condensation process only, heating Q can be written as Q +2 = LC + 2. (42) 8

9 Celal S. Konor Release. 3/2/08.5.c. Vertical ass flux equation for teporally- vertically-discrete syste Let us define, p as the deviation of respected variables due to horizontal advection physical processes at odels grid points. Note that a significant portion of is due to PL-top entrainent. Then we define ( F) = F( +,p + p, + ) F(, p, ). (43) Since F reains unchanged on surfaces, ( F) ust be copensated by the vertical advection, therefore, ( F) + ( t) F G p t F = 0, (44) where ( t) is the tie step. Fro (44), we obtain the vertical ass flux equation for the tie-discrete case = ( F) t t F F G p. (45) The solution of (45) requires iteration. During the iteration, p change following ( t) = ( ) ( p t) =, respectively. ( ), which is calculated fro + p + p, the PL-top pressure (= + ) reain unchanged Iteration continues until calculated becoes virtually zero. Then the vertical ass flux is deterined fro = ( p final p initial ) ( t ), where p final p initial ( p + p) are the pressure at the end at the beginning of the iteration, respectively. In the vertically discrete syste, the vertical ass flux equation (45) is applied at the odel interfaces. ( ) ust be consistent with the vertical finite difference ter in the discrete therodynaic equation given by (4e) 4g)..5.d. Moisture equation At the upper ost interface of the odel, the equation that predicts the assweighted water vapor ixing ratio is written as ( q) 2 t = ( qv) 2 q 2 C 2, (46) 9

10 Celal S. Konor Release. 3/2/08 where q is the water vapor ixing ration ( q) 2 q 2, (47a) 2, (47b) 2, (47c) ( qv) 2 q 2 v, (47d) ( q ) 2 q (47e) Within the free atosphere, the equation that predicts the ass-weighted water vapor ixing ratio is written as where ( q) + 2 t = ( qv) + 2 ( qv) + 2 q + 2 [ q + ] C + 2, (48) ( q) + 2 q , (49a) + 2 2( ) [ ], + 2 (49b) ( ) +2 2[ ) + ], (49c) q [ + + v + + v ], (49d) + 2 Definitions of ( q ) will be discussed in ore detail later in this text. For convenience in the early versions of the odel, we used ( q ) 2 q + 2( ) + q ( ) 2. With this definition the vertical oisture advection schee becoes a second-order centered finite differencing when is unifor is constant. Such a centered schee ay produce large dispersion errors it cannot accurately represent the PL-Free Atosphere ass exchange process. At the lower ost interface of the free atosphere, the equation that predicts the ass-weighted water vapor ixing ratio is written as 0

11 Celal S. Konor Release. 3/2/08 L + 2 q t = ( qv) L + 2 L + 2 ˆ q L + 2 ( ) gm L q L L + 2gM q L ( C) L+ 2, (50) where ( q) L + 2 L + 2 q L + 2 L, L, (5a) (5b) ( ) L +2 2 ( ), (5c) L ( qv) L + 2 q L + 2 L v L. (5d) In (50), we used upstrea treatent for exchange associated with M. The assweighted vertical ass flux at the lowest layer of the odel is given by ( q ) q L 2 ˆ L + 2 ( ) + q L + 2 L 2 ( ) L 2, where ˆ q L + 2 q L + 2 L ˆ q L + 2 r L + 2 if if ( ) gm L +2 < 0. (52) ( ) L +2 gm > 0 Following the arguent above, the expression for ( q ) L is odified to better represent the PL-Free Atosphere ass exchange process. This will be discussed later in this text..5.e. Vertical oisture fluxes To calculate the vertical oisture fluxes at the layers, we will rewrite (48) by oitting horizontal advection diabatic ters as ( q) +2 t = q + 2 [ + ( q ) ] (53) Following Hsu Arakawa s (990) positive-definite schee, we define the vertical ass fluxes if ( q ) is upward [ ( ) + ( ) ( ) 0 ] by + q q 2 2 ( + ) + q ( 2 q +2 ) ( + ) ˆ + ( q + 2 q + 32 ) for =,2,...,L (54)

12 Celal S. Konor Release. 3/2/08 ( q ) L We define the vertical ass fluxes if by ( q ) In (54) (56), q q 2 2 L + ˆ q L +2. (55) is downward [ ( ) ( ) ( ) + 0 ] ( ) q ( + 2 q 2 ) ( q ) q2. ˆ q 2 q 32 for = 2,3,...,L (56) = 29, (57a) ± ± (57b) ˆ ± ˆ ±, (57c) + = ˆ + 2 P +2 2 P +2 + q +2 q 2 2 for =,2,...,L, (57d) = ˆ 2 P 2 2 P 2 + q +2 q 2 2 for = 2,3,...,L, (57e) P + 2 q q +2 + q 2 +, (57f) ( + ) ( + ) + +, (57g) 2

13 Celal S. Konor Release. 3/2/08 ( ) ( ). (57h) In (57f), is an infinitesially sall positive constant..5.f. Correction step of oisture prediction In the tie discrete case, the solution of the vertical advection schee discussed above is subject to over under shooting errors. To iniize these errors, we eployed a procedure, which is based on the principle that the advection process cannot generate new axius inius in the field. Prior to the vertical advection, we locally deterine the upper lower bounds of advected values by { } { } ( q ax ) 2 = Max q 2,q 32 ( q in ) 2 = Min q 2,q 32 for ( q ax ) 2 = ( q in ) 2 = q 2 for ( ) > 0 < 0, (58a) ( q ax ) + 2 = Max q +2,q + 32 ( q ax ) + 2 = Max q +2,q 2 { } for ( ) > 0 + { } for ( ) < 0 for =, 2,, L, (58b) ( q in ) + 2 = Min q + 2,q +3 2 ( q in ) + 2 = Min q + 2,q 2 { } for ( ) + > 0 { } for ( ) < 0 for =, 2,, L. (58c) In (58b) (58c), q L+2 q L+2. At the lowest interface of the free atosphere, the estiated ax in values are ( q ax ) L + 2 ( q ax ) L + 2 ( q in ) L + 2 ( q in ) L + 2 { } for ( ) > 0 {,q L 2 } for ( ) < 0 = Max q L L +2,q L +2 = Max q L +2 L {,q L +2 } for > 0 {,q L 2 } for ( ) < 0 = Max q L + 2 = Max q L + 2, (58d). (58e) 3

14 Celal S. Konor Release. 3/2/08 In (58a-e), q + 2 (for = 0,, 2,, L) are pre-advection values. Then, after the advection, we apply corrections. Now let us first express q + 2 for = 0,, 2,, L as post-advection values. Then the correction process can be suarized as follows If ( ) L + 2 > 0, If q L +2 < ( q in ) L + 2 If ( ) > L 0: If q L +2 > ( q ax ) L + 2 q L +2 L q L +2 = ( q in ) L +2 L = q + 2 q L 2 = q L 2 q L +2 = ( q ax ) L + 2 ( q in ) [ L + 2 q L +2 ] L [ q L +2 ( q in ) L +2 ] L + 2 L +2 ( ) L +2 L L + 2 L 2 ( ) L 2 For = L, L 2,, if ( ) > 0, If q +2 q 2 = q q +2 = ( q ax ) + 2 > q ax where q L + 2 q L +2. For =,2,,L if ( ) > 0, If q 2 < q in + [ q +2 ( q in ) +2 ] + 2 q 2 = ( q in ) 2 2 q +2 = q + 2 ( q in ) 2 q 2 [ ] ( ) ( ) +2 where q L + 2 q L +2. for =,2,,L For =,2,,L if ( ) < 0, If q 2 q 2 = ( q ax ) 2 2 > q ax q +2 = q [ q 2 ( q ax ) 2 ] ( ) +2 4

15 Celal S. Konor Release. 3/2/08 where q L + 2 q L +2. For =,2,,L if ( ) < 0, If q +2 q 2 = q 2 ( q in ) +2 q q +2 = ( q in ) +2 < q in where q L + 2 q L +2. [ ] ( ) 2 During the correction process, no odification is ade to the values if ( q in ) + 2 q +2 q ax }. +2 {i.e. q + 2 = q +2.5.g. Moentu equation by The vertically discrete oentu equation applied to the odel layers is given v t + v v + v = ( p ) fk v for = 2,3,...,L, (59) where v is the horizontal velocity, f is the Coriolis paraeter, k is the unit vertical vector. The vertical advection of oentu is defined by v v v L 2 ( v 2 v ) 2 L ( v + v ) L v L + 2 v L wherev L +2 =v + f + v L,v L 32 60a +2 + v v 2 for = 2, 3,...,L ( ) + ( 2 v L v L )( ) L 2 for for v L +2 =v f v L+ 2,v L+ Currently, we are using v + v L v v L+..5.h. Horizontal pressure gradient force ( 60b), ( 60c) ( 60d) < 0, which is an extrapolation fro above, ( ) > 0, which is an extrapolation fro PL. 5

16 Celal S. Konor Release. 3/2/08 The first ter on the right h side of (59) is the pressure gradient force given by ( p ) = M +, (6) where the Montgoery potential is given by M +..5.i. Hydrostatic equation The vertically discrete hydrostatic equation is given by L = L+2 + ( L+ 2 L ) L+2 (62a) = + ( 2 ) + ( 2 ) for = L,L,...,. (62b) In (62a), L+2 is obtained by vertically integrating hydrostatic equation within the PL starting fro the Surface. L+2 is the predicted potential teperature at the lowest interface of the free atosphere, which is also expressed as + L+2 is the PL-top Exner function. The influence of the oisture on the geopotential height can be included in (62a) (62b) by replacing the potential teperature by the virtual potential teperature given by v ( q). The geopotential height at the interfaces can be calculated fro or 2 = + ( 2 ) for = L,L,...,. (63) +2 = ( +2 ) for = L,L,...,. (64) 6

17 Celal S. Konor Release. 3/2/08 2-Vertical discretization in the PL This section describes the vertical discretization in the PL. The PL consists of ultiple layers between the free atosphere of the odel Earth s surface. A shared coordinate surface, referred as PL-top, separates the free atosphere fro the PL. The height of the PL-top is predicted through a ass budget equation for the PL. The ass budget of the PL is priarily controlled by the PL-top entrainent horizontal ass convergence within the PL. Konor Arakawa (2000) discuss the rational behind the ulti-layer PL approach. The vertical discretization within the PL follows Arakawa Konor (996), which describe a siga vertical coordinate odel with a Charney-Phillips type vertical grid. 2.-Vertical coordinate in the PL by A siga type vertical coordinate is used in the PL portion of odel, which is given p p ( ) p S (65) with S = 0 < <. The constant PL ass is written by p = + ( ). p = p S S (66) In the odel, we first prescribe ( y, p) field between p S ( y) p T to be used in the deterination of the coordinate. Then we define ( y) to represent the initial PLtop. We then prescribe the function g( ) obtain the corresponding f ( ) as discussed in subsection.a. This gives us, with a prescribed, S. For (65) (66), we can obtain = ( p), respectively, for the PL. 7

18 Celal S. Konor Release. 3/2/ Vertical grid in the PL Lower Portion of the Vertical Grid L-/2 L L+/2 L+/2 L+ L+3/2 -/2 p p p p p θ q ~ θ q ~ θ L q L ~ L θ q ~ θ q ~ Q C Q C Q L R L (F ψ ) (F ψ ) K ψ Q R Q R (F ψ ) K ψ Free At PL +/2 p θ q ~ Q R M (F ψ ) K ψ S M+/2 p θ q ~ S =0 Q R (F ψ ) S 2.3. Vertically discrete equations in the PL 2.3.a. Continuity Vertical ass flux Equations The continuity equation within the PL is given by t = Here we ake following definitions M ( v) k ( ) k + PL k = L +. (67) ( p) p + 2 p 2 ( p) PL p S, (68) ( ) ( ) PL S where we used as vertical index. Fro (67) (68), we can obtain pressure tendency equations within the PL as 8

19 Celal S. Konor Release. 3/2/08 p +2 p S t t = t = t + ( v) k ( ) k + M k = L + + v k =L + k k ( ) +2 ( ). (69) The PL-top ass flux is deterined fro the entrainent/detrainent cuulus ass flux by ( ) = g E M ( ), (70) where negative E corresponds to detrainent. A liiting procedure is applied to [or ( ) ] to prevent the PL to get very deep or shallow. It is assued that the L +2 liiting procedure odifies E while M reains unchanged. The equation that deterines the vertical ass flux within the PL is written as ( ) + 2 = S + 2 ( S ) ( ) ( ) S M k =L + ( v) k k ( v) k ( ) k. k =L + (7) 2.3.b. Therodynaic Equation At the lowest interface of the free-atosphere (), the vertically discrete therodynaic equation is written by L + 2 L + 2 t L + 2 L v + L + 2 L + 2 = Q, (72) L + 2 where L + 2 L + 2 ˆ ( L + 2 L + 2 ) L + 2 gm + L ( L + 2 L + 2 )gm ( ) L } (73) + L + 2 L ˆ L + 2 L + 2 ˆ L L + 2 L + 2 if if ( ) L +2 gm < 0. (74) ( ) L +2 gm > 0 9

20 Celal S. Konor Release. 3/2/08 In (73) (74), we used upstrea treatent for the exchange associated with M. On the other h, at the Upper ost interface of the PL (L), the vertically discrete therodynaic equation is written by L + 2 t L + 2 L v L + ( = Q) L L+ 2 g L L+ 2 L+ 2 L L + 2 [ ( F ) F L+ L+ 2 ]+ G L+ 2, (75) where L L + 2 L L L + L + 2 L + 2 L ( ) L + + L + 2 ˆ ( L + 2 ) ( ) L + 2 gm.(76) L ( v) L + 2 L v L +2 L L + 2 v L + L v L + 2 ( 2 ) L + ( ) L L +3 2 L + 2. (77) In (76), we used upstrea treatent for the exchange associated M. Note that F G in (75) represent turbulent fluxes other PL processes in the PL. They should not be istaken by the functions used in the definition of the vertical coordinate in subsection.a.b. Within the PL, the vertically discrete therodynaic equation is written by ( v) t ( ) + 2 ( = Q) + 2 g [ + ( F ) ]+ G where F for = L +, L + 2,, M,(78) 20

21 Celal S. Konor Release. 3/2/08 ( ) p +3 2 p 2 2p + 2 ( ) for = L +, L + 2,, M, (79) ( v) v + 2 [ + + ] [ ( v ) ( ) ] for = L +, L + 2,, M, (80) At the lowest interface of the PL, the vertically discrete therodynaic equation is written by M+ 2 t where + ( v) M+ 2 M+ 2 ( = Q) M+ 2 M+ 2 g [ ]+ G ( F ) M+ 2 ( F ) M M+ 2 ( v) M + 2 v M+ 2 v M+2 v M M+ 2 2 M M+ 2, (8). (82) 2.3.c. Moisture Equation The oisture equation applied to the upper ost interface of the PL (L) can be given by L + 2 t q L qv L + L L + 2 ( q ) L + ˆ q L + 2 ( ) L + 2 gm + q L L + 2gM = ( C) L + 2 g L L + 2 ( F q ) F L + q L ( G q ), (83) L + 2 where L ( qv) L+ 2 L q L+ 2 v L+ 2, (84a) 2

22 Celal S. Konor Release. 3/2/08 ( q ) q L + L +3 2 ( ) L + if ( ) > 0 L + L ( q ) L + q L + 2 ( ) L + if ( ) L + < 0 (84b) ˆ q L + 2 q L + 2 L ˆ q L + 2 q L + 2 if if ( ) L +2 gm < 0. (84c) ( ) gm L +2 > 0 Note that F q G q in (83) represent turbulent fluxes additional PL processes in the PL. They are different fro the functions used in the definition of the vertical L coordinate in subsection.a.b. The predicted q L+2 is further corrected using a procedure siilar to the one described for the free atosphere. We will discuss this procedure later in this text. The oisture equation applied to the interfaces within the PL can be given by ( q) + 2 t where + ( qv) q + 2 = ( C) + 2 [ + ( q ) ] [ F + ( q ) ]+ ( G q ), (85) + 2 g + 2 F q ( qv) + 2 q +2 [ ] (86a) v v + 2 ( q ) q + 2 ( ) if ( ) > 0 ( q ) q 2 ( ) if ( ) < 0. (86b) In (86b), { } ( 2 ) for = L +,L + 2,,M. The predicted q is further corrected using a procedure siilar to the one described for the free atosphere. We will discuss this procedure later in this text. The oisture equation applied to the lowest interface of the PL can be given by ( q) M + 2 t + ( qv) M q M+ 2 M+ 2 q M+2 M 22

23 Celal S. Konor Release. 3/2/08 = ( C) M + 2 g F q M + 2 F M + 2 ( q ) M + ( G q ) (87) M+ 2 where ( qv) M+ 2 q M+ 2 v M (88a) In (88b), discussed below. ( q ) q M M + 2 ( ) M if ( ) > 0 ( q ) q M M 2 ( ) M if ( ) < 0. (88b) M ( 2 ). A correction procedure on the predicted q M2 M+2 is 2.3.d. Correction step of oisture prediction The estiated ax in values at the upper ost level of the PL are L ( q ax ) L + 2 L ( q ax ) L + 2 L ( q in ) L + 2 L ( q in ) L + 2 { } for ( ) > 0 { L,q L +2 } for ( ) < 0 L = Max q L +3 2,q L + 2 = Max q L + 2 { } for ( ) > 0 { L,q L +2 } for ( ) < 0 = Max q L +3 2,q L + 2 = Max q L +2 The estiated ax in values at the interfaces within the PL are, (89a). (89b) ( q ax ) + 2 = Max q +2,q + 32 ( q ax ) + 2 = Max q +2,q 2 { } for ( ) > 0 + { } for ( ) < 0 for = L +,L + 2,,M, (89c) ( q in ) + 2 = Min q +2,q + 32 ( q in ) + 2 = Min q +2,q 2 { } for ( ) > 0 + { } for ( ) < 0 for = L +,L + 2,,M. (89d) 23

24 Celal S. Konor Release. 3/2/08 L In (89c) (89d), q L+2 q L+2. At the lower boundary, the estiated ax in values are given by ( q ax ) M+2 = ( q in ) M+2 = q M +2 for { } { } ( q ax ) M+2 = Max q M + 2,q M 2 ( q in ) M+2 = Min q M +2,q M 2 for M > 0 ( ) M < 0 (89e) If ( ) > M 0: If q M + 2 q M 2 = q M2 M + 2 q M + 2 = ( q ax ) M+2 > q ax [ ( q in ) M+2 ] + q M+2 For = M,M 2,,L + if ( ) > 0, M + 2 M 2 If q +2 q 2 = q q + 2 = ( q ax ) +2 > q ax [ ( q in ) +2 ] + q where q L + 2 q L +2. If ( ) L + 2 > 0 : L If q L +2 L > ( q ax ) L + 2 q L +2 L q L +2 = q L +2 + q L + 2 L = ( q ax ) L +2 [ ( q in ) L +2 ] L L + 2 L + 2 L +2 If ( ) L + 2 > 0, If q L +2 < ( q in ) L + 2 q L +2 L q L +2 = ( q in ) L +2 L = q + 2 ( q in ) [ L + 2 q L +2 ] L + 2 L +2 L L + 2 For = L +,,M 2,M if ( ) > 0, 24

25 Celal S. Konor Release. 3/2/08 If q +2 q + 2 = q q 2 = ( q ax ) 2 < q in [ ( q in ) 2 ] + q where q L + 2 q L +2. For = L +,L + 2,,M if If q +2 < q in < 0, [ ] q 2 = q 2 ( q in ) +2 q q + 2 = ( q in ) + 2 where q L + 2 q L +2. For = L +,L + 2,,M if ( ) < 0, If q 2 < q in [ ] q + 2 = q + 2 ( q in ) 2 q q 2 = ( q in ) e. Moentu Equation The vertically discrete oentu equation applied to the layers in the PL is given by v L + t v t + v L + v L + + v + v v + v L + = ( p ) f k v L + L + = ( p ) +f k v g F v L + g ( L G v ) L + { } + G v ( F v ) +2 ( F v ) 2 ( 76a) = L + 2,...,M ( 76b), 25

26 Celal S. Konor Release. 3/2/08 where the vertical oentu advection is v v v L + M ( L + 2 v v L + 2 L +) 2 ( v + v ) 2 ( v M v M ) M + v L + v L + 2 L + 32 ( ) ( v v )( ) 2 for = L +,..,M ( 77a). ( 77b) M 2 77c In (77a), we forally define v L +2 =v + f + v L,v L for < 0, which ust be ( ) > 0, which for an extrapolation fro above, v L +2 =v f v L+ 2,v L+ ust be an extrapolation fro PL. Currently, we are using v + v L v v L f. Horizontal pressure gradient force On the right h side (77a) (77b), the vertically discrete pressure gradient force is given by ( p ) = + [ ] ( p + 2 p 2 ) p p for = L +,L + 2,,M (78) 2.3.g. Hydrostatic equation The geopotential height within the PL is deterined by vertically suing the hydrostatic equation starting fro the surface where M+2 S is prescribed. The vertically discrete hydrostatic equation is given by M = M+2 + ( p M+ 2 p M2 ) 2p M+2 M+ 2, (79a) M+2 = + + p + 32 p 2 2p for = M,M 2,,L +. (79b) At the top of the PL, the geopotential height is calculated fro 26

27 Celal S. Konor Release. 3/2/08 L+2 = L+ + ( p L+ 32p L+ 2 ) L 2p L+2 L+2, (79c) L+ 2 where L+2. At the interfaces of layers, the geopotential height can be calculated fro or 2 = + ( p p +2 2) 2p 2 2 for = M,M,,L + 2 (79d) = p +2 p 2 2p for = M,M 2,,L + (79e) 27