Some Sub-problems - BOUNDED DERIVATIVE METHOD Introduction :

BOUNDED DERIVATIVE METHOD Introduction :

12. Some Sub-problems

d dtA_(λ)⎤

⎥⎦C

∫

₀^λ^max^K^{(λ , λ′) m}^B^(λ′⁾^d^λ′

i.e.^⎧⎨

⎩A (ⁱ)_t_{+ ∆t}−^A(i)_t^⎫^⎬_⎭=

∑

j = i j = i_max

(λ^,λ′)m_B(λ′) ∆^t

7.3(15) ii) Take small arbitrary values for m_B(j) and ∆^t^{; for} convenience of numerical calculation, let m_B(j)∆t

= Constant . By using the set of equations given in the original text (Arakawa and Schubert,1974), find the value of

K(i ,j)=^⎧^⎨_⎩A (i)t +∆^t− A (i)t

⎫⎬

⎭

/

^m^B⁽^j^{) ∆}^t 7.3(16) for each value of j i.e. the change in cloud work fu nc t i o n o f th e i-th sub-ensemble due to modification of the large-scale environment by j-th sub-ensemble, per unit mass flux m_B(j), per unit time.

9. Calculation of F(λ⁾ i) ⎡

⎢⎣ d d tA (λ)⎤⎥

⎦LS

=F (λ)

F_(i)=^⎧⎨

⎩A(ⁱ)t + ∆ t−A (i)t

⎫⎬

⎭

/

^∆^t 7.3(17) ii) By using the set of equations given in the original text (Arakawa and Schubert, 1974), calculate F(i) , the effect of large-scale processes on large-scale environment, in particular, in changing A (i); ∆t 30 minutes is acceptable.

10. Calculation of m_B (λ′)

In integral form, we have to solve Fredholm’s Integral Equation of first kind

∫

_{λ′ = 0}^λ^max^K^{(λ , λ′}⁾^m^B^{(λ′) d}^λ^′+^F^(λ)=0

under the condition that m_B(λ′) is positive. In passing, it may be mentioned that there are two kinds of Fredholm’s Integral euqations :

First Kind :

∫

_a^b^K^(x^,^y^{)f (y)}^d^y⁼^g^(x) ^7.3(18)

Second Kind :

∫

_a^b^K^(x^,^y^{)f (y)}^d^y⁼^{µ g}⁽^x⁾^+f(x) 7.3(19) K is known and is called Kernel. f i s unknown and is to be found; µ^andg are known.

11. Schematic Diagram

Fig. 7.3 (1) is a schematic diagram of a part of the parameterization Scheme. T__

and q_

denote the values of temperature and specific humidity of the synoptic-scale environment at time t. From these parameters, one can get λ^andA (λ^).

i ) E f fe c t o f L a r g e - s c a l e Pr o c e s s e s o n Environment :

Choose a small time interval ∆ t′ and calculate changes in temperature and specific humidity and get T__

and q_

at time t + ∆ t". From these, calculate λ and A⁽λ^{) at time}t +∆ t′ . The time rate of change of A (λ^{) gives}F (λ^{) , i.e.}F(i).

ii) Effect of Cloud-Cloud Interaction on Environment :

Si mi l a rly, c al cul at e c h ange s in temperature and specific humidity in small time interval ∆ t ′′ due to cloud-cloud interaction. For the changed values of T__

and q_

, calculate λ and A (λ) at time t +∆ t ′′ . From the time rate change of A (λ) due to cloud-cloud interaction, find K (λ , λ′),i.e.K (i , j) .

iii) For these given values of F ( i ) and K(i , j ), solve Fredholm’s integral equation :

∫

_λ′^λ^max^K⁽^λ^,^λ′)m^B(λ′) d λ′+F(λ)=0 or

∑

^⎡_⎣^K^{(i , j)m}^B^(j)⎤_⎦

j=¹ i_max

+F (i) = 0 7.3(20)

In other words, for a particular i, given F(i) and K ( i , j ) for j = 1,2,3,...,i_max,find the set of values m_B ( j ) for j = 1,2,3,...., i_max

12. Some Sub-problems

We sh a ll now out li ne some of t h e sub-problems of the Arakawa-Schubert Scheme, also using some numbers in place of symbols i and j when useful for fixing the ideas.

Sub-problem I :

Given distribution of T__

and q^_ , h o w to calculate λ_i^?

Ans.

i) Specify sub-ensemble λi by pressure level of zero buoyancy, p^

i. This can be calculated from given distribution of T__

and q^_ . We may call this as pressure of cloud-top of sub-ensemble λi. Equating the level of zero buoyancy to cloud top level implies that detrainment takes place in a thin layer around this level, because level of zero buyoyancy is the level of maximum upward velocity.

ii) To get the correct value of λi for a given p^

i, adopt the following iterative process : a) Assume a guess value of λi

b) Using this guess value of λi and the given vertical distribution of T

__and q^_, calculate the

level of zero buoyancy. Let it come as p^^ _i c) If p^^

i <p^

i , increase λi by a small amount.

If p^^

i >p^

i , decrease λi by a small amount.

(Larger value of λ will give smaller height and larger pressure value of cloud-top).

d) Go to step (b) and calculate new value of p^^

i for new value of λi . Go to step (c).

e) Use this iterative process till you get proper λi such that ⎪^p^{^^}_i = p^

i⎪ for all practical purposes i.e. ⎪^p^{^}i − p^^ _i ⎪ is smaller than some specified FIG. 7.3(1) : Schematic diagram for determining K (λ ,λ′ )^,F(λ^{) &}m_B(λ′ ) (Asnani, 1993).

small quantity. This gives mutually compatible va lue s of λi a n d p^ _i for any given vertical distribution of T⁻ and q_

Sub-problem II : Given ⁻T , q_

, λ and p how to get A₍λ)?

Ans.

A₍λ)=

∫

z_t

η (z , λ) g

T−( z) ^⎧^⎨^⎩T_vc₍z , λ) − T−v(z)^⎫^⎬_⎭ d z Lord and Arakawa (1980) chose cloud-top pressure to denote a cloud sub-ensemble. For each value of cloud-top, they calculated corresponding val u e of λ. Field observations directly give cloud-tops and vertical distribution of and q bar and not λ . For this type of data set, it is possible to calculate λ corresponding to each cloud -top. Lord and Arakawa (1980) chose the following 17 values for cloud-top pressure (mb, hPa) to identify 17 cloud sub-ensembles :

100, 150,200,250, 300, 350, 400, 450, 500, 550, 600, 650, 700, 750, 800, 850, 912.5 mb (hPa).

For all these cloud-tops, the base of the cloud was taken to be 950mb. Corresponding to these 17 values of cloud-tops, they calculated λ and A(λ) for different, vertical distributions of ⁻T and q^_. For observations of p^ ,T− and q^_ , they took 7 different locations and synoptic conditions in the tropics and sub-tropics. The 7 locations were :

i) The Marshall Islands data set from 15 April to 22 July 1956 (Yanai et al. 1973, 1976).

ii) VIMHEX data set over north-central Venezuela from 22 May to 6 Sep 1972 (Betts and Miller, 1975).

iii) GATE data set from 31 August to 18 September 1974 (Thompson et al., 1979).

iv) AMTEX data set from 14 to 28 February 1974 and 14 to 28 February 1975 (Nitta,1976).

v) Mean West Indies Tropical Sounding for hurricane season July to October for 10-year period 1946-1955 (Jordan,1958).

vi) Composited Northwest Pacific Typhoon data set from mean soundings for 10-year period 1961-70, given by Frank(1977).

vii) Composited West Indies Hurricane data set based on observations at coastal and island stations

in and near the Gulf of Mexico and the Caribbean Sea (Nunez and Gray, 1977) for the 14-year period 1961-74.

For each of these data sets, Lord and Arakawa drew the graph of A(λ) versus cloud-top pressure. All the graphs showed great similarity, suggesting something like a universal relationship between A(λ) and λ, the latter being represented by cloud-top pressure p^^. This type of universal r e l a t i o n s h i p g iv e s a g o o d su p p o r t to Arakawa-Schubert Scheme of parameterization.

We may here remind ourselves that while A(λ^{) is} calculated from formula 7.3 (11) from the vertical distribution of T− and q_

of the environment, the Arakawa-Schubert Scheme of Parameterization further assumes quasi-stationarity of A(λ) over small intervals of time of the order of 30 minutes.

This additional property of universal relationship between λ and A(λ) in situations widely-separated both in space and time, lend some additional support for taking A(λ) as a basic parameter in the parameterization scheme.

Fig. 7.3 (2) shows the relationship between fractional entrainment rate λ and A(λ) for the GATE dataset. For the other 6 data sets analysed by Lord and Arakawa (1980), the distributions were similar to this in many respects. Fig 7.3 (3) shows the relationship between fractional entrainment rate and cloud-top pressure for the GATE data. This diagram enables the transformation of x-axis from cloud-top pressure to fractional entrainment rate and vice-versa. Fig. 7.3 (4) shows the distribution of cloud-work function A(λ) versus cloud-top pressure for the 4 data sets of Marshall Islands, VIMHEX,GATE and AMTEX, taken from Lord and Arakawa(1980).

Sub-problem III : Given T− , q_

,λ , p^ (λ),^andA(λ) ; how to get K (λ^,λ′ )andF (λ)^?

Ans.

i) d

d tA(λ)=

∫

_λ^λ_′^max_{= 0}^K^{(λ , λ}^′^{) m}^B^(λ^′⁾^d^λ^{′+F (λ)}

ii) Specify all possible discrete values of λ . As already stated, Lord and Arakawa (1980) chose 17 values for p^ . Corresponding to these 17 values of p^, they calculated 17 values of λ and A(λ) for

given vertical distribution of T__

and q_

. The set of 17 values completely defines the cloud-ensemble.

Sub-ensemble λ≡ i , sub-ensemble λ′==j ;i, j=1,2,3,...,17.

iii) Choose a particular value of i, say i = 3.

d t A ₍₃₎₌

∑

j = 1 j=17

(3 , j)mB(j)+F (3)

iv) For the moment, consider changes in A(3) due to cloud-cloud interaction only.

a) Choose a small time interval ∆ t ′′.

A (3)t + ∆ t′′−A (3)t=

∑

j= 1 j = 17

(3 , j)m_B_{(j)∆ t ′′}

b) Take j =1 and an arbitrarily chosen small amount of mass flux at cloud base m_B′′ (1), of cloud sub-ensemble 1.

Calculate ^⎧⎨

⎩ A (3)_t₊_∆t_′′−A (3)_t^⎫^⎬_⎭ due to m_B′′

(1). Then

K(3 , 1)=^⎧^⎨_⎩A(3)t + ∆t ′′−A (3)t

⎫⎬

⎭

/

^m^B^{′′ (}¹) ∆ t ′′ 7.3(21) c) Similarly, take j = 2,3,4,...,17 and calculate

K(3,2),K(3,3),....,K(3,17).

v) Now consider change in A(3) due to large-scale processes only. Choose a small time interval ∆^t′.

F(3)=^⎧^⎨_⎩A (3)t₊_{∆ t}′−^A(3)t

⎫⎬

⎭

/

^∆^t′ 7.3(22) Sub-problem IV :

Given ⁻T , q_

, λ , p^ (3), A(3), F(3), K(3,1),

K(3,2), K(3,3),..., K(3,17); how to get m_B(1), m_B(2), m_B(3),...,m_B(17)?

Ans.

i) Using the assumption of quasi-stationarity of A (3) over small interval of time ( 30 minutes), we have to solve the equation

∑

j = 1 j = 17

(^{3 , j})mB(j)+F (3)=0 7.3(23)

∫

_{λ′ = 0}^λ^max ^K(λ , λ′ ) mB(λ′) d λ′+F(λ)=0 Under the condition that m_B(λ′) is positive.

K(3,1) m_B(1) + K(3,2) m_B (2) +K(3,3) m_B(3) FIG. 7.3(2) : Relationship between fractional entrainment

rate λ (10⁻² km⁻¹) and cloud work function A(λ) (J kg⁻¹) for GATE data set (Lord & Arakawa, 1980; Asnani, 1993).

FIG. 7.3(3) : Relationship between fractional entrainment rate λ ⁽¹⁰⁻²^km⁻¹) and cloud top pressure p^ (mb) for GATE data set (Lord & Arakawa, 1980; Asnani, 1993).

FIG. 7.3(4) : Relationship between cloud top pressure p^

(mb) and cloud work function A (λ^{) (J kg}⁻¹) for 4 data sets. Mean values and one standard deviation from mean value are shown on the two sides of the mean value. (Lord and Arakawa, 1980; Asnani, 1993).

+...+ K(3,17) m_B (17) + F(3) = 0

When m_B(λ′) is zero or negative, that cloud sub-ensemble is supposed not to exist.

ii) In this equation, K(3,1), K(3,2), K(3,3),...., K(3,17) and F(3) are known. But the 17 quantities m_B(1), m_B(2), m_B(3),....,m_B(17) are unknown. To get these 17 unknowns, we write 17 linear inhomogeneous equations in these unknowns : K(1,1) m_B(1) + K (1,2) m_B(2) + K(1,3) m_B(3) + ...+ K(1,17) m_B(17) + F(1) = 0

K (2,1) m_B (1) + K(2,2) m_B (2) + K (2,3) m_B(3) + ...+K(2,17) m_B(17) + F(2) = 0

. . . .

K(17,1) m_B(1) + K(17,2) m_B(2) + K(17,3) m_B(3)+

...+ K(17,17) m_B(17) + F(17) =0 7.3(24).

iii) These 17 linear equations in 17 unknowns are solved by standard methods.

iv) Lord (1982) suggestted simplex linear programming algorithm for solving these 17 linear inhomogeneous equations subject to the condition that m_B(j) are non-negative. This algorithm is further explained in the paper by Lord et al. (1982), which forms part IV of the Arakawa-Schubert Scheme.

Sub-problem V : Given _T^__ , q_

, λ , p^ ( λ ) , A ( λ ) , K(λ, λ′) , F (λ) and m_B(λ′) ; how to get the rate of precipitation and the rates of change of T__

and q_ ? Ans.

i) Obtaining of m_B(λ′) elements is a major step which has been explained in Sub-problem IV.

After m_B(λ′) elements are determined, the calculations for the rate of precipitation and the rates of change of T__

and q_

of the environment are relatively simple and straightforward. The steps for these calculations are explained by Lord (1982) and Lord et al. (1982). Readers may refer to these original papers for details of calculation. In these papers, the authors have also compared their model results with observations during Phase III of GATE, 1-18 Sep ’74.

ii) The aim of presenting Arakawa-Schubert Scheme here has been to make the outline of their approach clear to the reader, as far as possible. The explanation of the scheme is otherwise spread over

4 parts between 1974 and 1982.

iii) Quantitative precipitation forecasting (QPF) is major problem in tropical forecasting. The results of QPF for GATE area presented by Lord (1982) are quite impressive. The technique deserved to be tested for more occasions in different tropical locations. In the process of testing, some simplifications may also suggest themselves for adoption.

For Indian region, calculations of Cloud Work Function had been presented by Rama Varma Raja (1994, 1996, 1999).

Combined Updraft-Downdraft Model :

Arakawa-Schubert model described above does not include convective downdrafts, which are important components of tropical convective cloud systems; convective updrafts are, however, included explicitly.

Strength of updrafts in these convective cloud systems is intimately related to their tilt in the vertical, the height of cloud tops and the amount of rainwater in the clouds. Cheng (1989 a,b) emphasized the role of downdrafts and presented a combined updraft-downdraft spectral cumulus ensemble model, which can be incorporated into the Arakawa-Schubert cumulus parameterization scheme given above.

Cheng and Yanai (1989) further utilized this combined updraft-downdraft model to study the effects of meso-scale convective system on the heat and moisture budgets of larger-scale tropical cloud clusters, using the GATE Phase III data. They concluded that the inclusion of convective downdrafts resulted in warming and drying in the upper troposphere, and cooling and moistening in the lower troposphere.

Cheng and Arakawa (1990) described in detail, the incorporation of Cheng’s (1989 a,b) model into Arakawa-Schubert’s original model.

This incorporation necessitated slight modification in the definition of cloud work function originally given by Arakawa and Schubert (1974).

However, they found that this revision made no significant difference in the normalised cloud work function (Cheng and Arakawa, 1992).

Cheng and Arakawa (1992) also made semi-prognostic tests (one-step predictions) with updraft-only model and updraft-downdraft model in the UCLA General Circulation model, using data set of GATE Phase III.

They came to the following conclusions : i) Both the models predict cumulus heating profiles which agree amongst themselves and also with the actual atmospheric conditions.

ii ) The updraft-only model tends to over-estimate the cumulus drying rates throughout the entire cloud layer. On the other hand, the updraft-downdraft model predicts the cumulus drying effects reasonably well.

iii) With their computer code in use, the inclusion of the downdrafts slowed down the entire General Circulation computation by a factor of 2.8 Further work done on Arakawa-Schubert Scheme of Cumulus Parameterization

In a series of three papers published in J.

Atmos. Sci., 1 June 1989, Cheng and Yanai have emphasized that a parameterization scheme should involve not only thermodynamic features as done in the Arakawa-Schubert scheme, but also dynamic features like Vertical Wind Shear. The three papers are :

(i) Effects of downdrafts and Mesoscale convective organization on the heat and moisture budgets of tropical cloud clusters. Part I: A diagnostic cumulus ensemble model (M.D. Cheng, pp. 1517-1538).

(ii) Effects of downdrafts and Mesoscale convective organization on the heat and moisture budgets of tropical cloud clusters. Part II: Effects of convective-scale downdrafts (M.D. Cheng, pp.

1540-1564).

(iii) Effects of downdrafts and Mesoscale convective organization on the heat and moisture budgets of tropical cloud clusters. Part III: Effects of Mesoscale convective organization (M.D. Cheng and M. Yanai, pp. 1566-1588).

Th e y h a v e p re s e n t e d t h i s re v is ed A ra k a w a -S c h u b e rt mo d e l a n d c a l l ed i t Updraft-downdraft Model. The main features of this model are summarized below:

(i) It is a diagnostic model.

(ii) It gives updraft model and downdraft model.

(iii) Gives the formula for tilting angle of the updraft in terms of horizontal and vertical velocity components of the updraft.

u_c = horizontal component of velocity of updraft air relative to the cloud.

v_c = 0

w_c = vertical component of updraft velocity relative to the cloud.

Then, tan _{θ =}^u^c W_c

where θ is the angle between the updraft and the vertical direction.

(iv) Horizontal and vertical components of velocity of rain drops u_r and w_r are given by

u_r = u_c w_r= w_c-V_t

where V_t is the mean terminal fall velocity of raindrop given by Soong and Ogura (1973, J.

Atmos. Sci., 30, 879-893):

Vt = 36.34 (ρ^ q_r)^0.1364 ⎛

⎜⎝ ρo

ρ^

⎞⎟

⎠

1⁄²

ms⁻¹ q _r = rainwater-mixing ratio ^ρ = density of updraft air

ρ_o = density of air at ground level

(v) The updraft tilting angle can be interpreted as the angle required to maintain updraft buoyancy against loading effect of rainwater.

For each sub-ensemble, it is assumed that there is statistically steady updraft. Tilting angle is considered to be a constant for each sub-ensemble.

FIG. 7.3(5) : The up draft tilting angles of various types of clouds obtained by Scheme A (solid), Scheme (B) (dashed) and Scheme C (long dashed). (Ming - Dean Cheng., 1989).

The updraft parameters u_c and w_c are obtained from Arakawa-Schubert Scheme of heat and moisture budgets.

(vi) The tilting updraft model is tested against the data of GARP (Phase III). The horizontal distribution of θ is nearly uniform when there is scattered convection; however, when there is organized cumulus convection, the updraft-tilting angle shows local maximum.

(vii) Fig. 7.3(5) gives the updraft-tilting angle θ for various types of clouds. The tilting angle increases with the depth of the cloud having the same cloud base near 960 mb (hPa). The tilting angle is less than 3^o for clouds having tops at 700 mb (hPa). For clouds having tops near 200 mb (hPa), the tilting angle at the top is of the order of 30^o.

For details of schemes A, B, and C mentioned in Fig. 7.3(5), the reader may refer to Part I of the three-paper series in J. Atmos. Sci., 1989, pp. 1517-1538. However, it is sufficient for our purpose to note that the schemes A, B, and C are simplified versions with a constant tilting angle for each ensemble updraft in the vertical. The results for the three schemes are not very different.

(viii) Downdraft model is also presented.

(ix) In Part II, Cheng presents the effect of convective-scale downdrafts on heat and moisture budget of tropical cloud cluster.

(x) In Part III, Cheng and Yanai present the effect of meso-scale convective organization on heat and moisture budgets of tropical cloud clusters.

(xi) The three papers together quantify relationship between updraft tilting angle, thermodynamic properties of the air mass and wind shear.

(xii) Anvil effects are : Warming and drying in the upper troposphere (of secondary importance, but not negligible). Correspondingly, there is cooling and moistening in the lower troposphere.

(xiii) Local maxima of the tilting angle appear well before the organized precipitation patterns, associated with squall clusters, and can be detected by radar; this implies the existence of thermodynamically preferred regions for the formation of cloud clusters.

(xiv) Larger tilting angle generally gives larger downdraft mass flux relative to the updraft mass flux.

(xv) Degree of cloud organization i s

related to the vertical wind shear. When vertical wind shear is large, then squall-clusters are likely to occur; when vertical wind shear is moderate,

"non-squall clusters" are more likely.

(xvi) Large vertical wind shear favors large t i l t i n g a n g l e a n d d e ep cu mu l u s in thermodynamically preferred regions. This proves c o u p l i n g bet we en t h e w ind fi el d and t h e thermodynamic field.

(xvii) The tilting angle and the cloud work function are negatively correlated in time; the tilting angle usually increases during the periods when the mass flux of the deep clouds associated with squall clusters is diagnosed.

(xviii) Occurrence of cloud clusters generally follows a long-term build-up of the cloud work function and the vertical wind shear. Short-term fluctuations are interpreted as the result of development and decay of organized cumulus convection.

(xix) The meso-scale organization of cumulus convection is a consequence of interaction between cumulus clouds and the environment under the influence of vertical wind shear. Dynamic parameters such as low-level wind shear should be taken into consideration in future cumulus parameterization schemes.

Inclusion of Updraft-Downdraft Phenomena in A r a k a w a - S c h u b e r t S c h e me o f Cu mu lu s Parameterization

In an International Symposium held in Indian Institute of Tropical Meteorology, Pune, India during 1992, Cheng and Arakawa presented the results of numerical experiments in General Circulation Models in which Cheng and Yanai’s (1989, J. Atmos. Sci., 1 June, pp. 1517-1588) t h r e e -se r i es pa pers w ere i n cl uded in A ra k a w a -S c h u b e rt S c h eme o f cu mu l u s parameterization. They brought out the following points:

(i) In the original Arakawa-Schubert scheme, the cumulus ensemble model did not inclu de convective downdrafts, which are important components in tropical convective systems.

(i i ) D o w n d r a ft s w h i c h i n v a r i a b ly accompany the updrafts inside the cumulus clouds, tend to decrease the cumulus heating and drying above the cloud base through a reduction of the subsidence between cumulus clouds.

(iii) The outflow from downdrafts below the cloud base may also significantly modify the thermodynamic properties of the sub-cloud layer.

(iv) Incorporation of the downdraft effects is a n o b v i o u s i mp ro v e m e n t i n t h e o ri g in a l A ra k a w a -S c h u b e rt s c h eme o f cu mu l u s parameterization.

(v) Cheng’s (1989 Part I, J. Atmos. Sci., pp.

1517-1538) model of combined updraft-downdraft spectral cumulus ensemble can be incorporated into the original Arakawa-Schubert scheme as outlined below:

(a) The rainwater generated in the updraft is assumed to fall partly inside and partly outside of updraft.

(b) The mean tilting angle determines this partitioning of the falling rain water.

(c) This mean tilting angle is estimated by considering stable statistically-steady states with random perturbations on the cloud-scale horizontal velocity.

( d ) T h e ve rti ca l ve lo c it y and t h e t h ermodyna mical prop erties of th e associated downdraft are then calculated considering the effects of rainwater loading and evaporation.

(vi) The updraft-downdraft model of Cheng (1989, Parts I & II, 1 June, pp. 1517-1564) was incorporated in the original Arakawa-Schubert scheme and tested diagnostically using a dataset for GATE Phase III. The results of this testing exercise are summarized below:

(a) The updraft-downdraft model predicts the cumulus drying rates reasonably well.

(b) The updraft-downdraft model performs better than the updraft-only model.

(d) The computer time requirement for this combined Arakawa-Schubert-Cheng model was about twice the time required for Arakawa-Schubert model. More experiments are required to reduce the computer time requirement. There is no doubt that the introduction of dynamics of downdrafts in the thermodynamic model of Arakawa-Schubert scheme is essential.

Xu, Arakawa, and Krueger (1992, J.

Atmos. Sci., 2402-2420) used two-dimensional UCLA cumulus ensemble model (CEM), covering

a large horizontal area with a sufficiently small horizontal grid size. They performed a number of simulation experiments to study the macroscopic behavior of cumulus convection under a variety of different large-scale and underlying surface c o n d i t i o n s . Th e y c a m e t o t h e fo l lo w in g conclusions:

(i) In all simulations, cumulus activity is

In document Tropical Meteorology by G.C. Asnani - Initialization and Parameterization (Page 41-49)