UNIFIED MODEL DOCUMENTATION PAPER 27

CONVECTION SCHEME

Version 3

D. Gregory, P.Inness and Julie M. Gregory

11 March 1999

Model version 4.4

Climate Research Meteorological Office

London Road BRACKNELL

Berkshire RG12 2SY

United Kingdom

(C) Crown Copyright 1999

This document has not been published. Permission to quote from it must be obtained fromthe Head of Climate Prediction at the above address.

Modification Record

DocumentVersion

Author Description

(Unnumbered) D.Gregory

2 D.Gregory:P.Inness:

Part 2 added August 1995 for version 3APart 3 added September 1995Part numbers in part 1 renumbered from 2 to 1.2etc. in contents list, and where found in text.

3 J. Gregory Corrections added to part 1 . Part numbersintegrated across document. Part 4 - anvil couldamount, new chapter.

2

OUTLINE

(1.1) INTRODUCTION

(1.2) CLOUD MODEL

(a) Definition of the bulk cloud model

(b) Discretised equations

(c) Entrainment and mixing detrainment

(1.3) INITIAL ‘DRY’ ASCENT

(1.4) CONDENSATION

(1.5) CALCULATION OF INITIAL MASS FLUX (CLOSURE)

(1.6) FORCED DETRAINMENT

(1.7) TERMINATION OF CONVECTION

(1.8) PRECIPITATION PROCESSES

(1.9) CONVECTIVE CLOUD BASE, TOP AND AMOUNT

(1.10) IMPACT ON ENVIRONMENT

(1.11) CHANGE OF PHASE AND EVAPORATION OF FALLING PRECIPITATION

(1.12) ENERGY CONSERVATION AND UPDATING OF MODEL VARIABLES

REFERENCES

LIST OF FIGURES

3

(1) INTRODUCTION

This paper describes a penetrative mass flux convection scheme used in the UK Meteorological

Office Unified Model. The scheme was originally devised in terms of sigma co-ordinates by Lyne

and Rowntree (1976) using GATE data, while Gregory and Rowntree (1990) provides a detailed

description of the scheme and documents its performance in single column model tests using a

variety of observational data and in a four year integration of the 11-layer atmospheric general

circulation model previously used for climate research at the UK Meteorological Office prior to the

introduction of the Unified Model system in 1990. The scheme was also employed in the 15-layer

limited area forecast model while the 15-layer global forecast model employed an earlier version

described by Slingo (1985). (The main differences are in the treatment of precipitation processes

and the detrainment of cloud air upon the termination of convection.)

The scheme described here is identical to that used in the 11-layer climate model but is now

described in terms of pressure rather than sigma co-ordinates as this is more appropriate to the

hybrid co-ordinates used in the unified model.

The scheme is applicable to moist convection of all types (shallow, deep, mid-level) together with

dry convection, and uses a single cloud model, based around parcel theory modified by

entrainment and detrainment to represent an ensemble of convective clouds, each of differing

characteristics and terminating at different levels. Hence the parcel characteristics (temperature,

mixing ratio, mass flux) calculated by the cloud model represent averages over the entire

ensemble. For a column of the atmosphere, working from the bottom upward, each layer of the

model is tested until one is found which, with a slight excess buoyancy s (0.2K in the standard

scheme, i.e that caused by a temperature excess of 0.2K), is still buoyant by more than a lower

limit b (set to 0.2K in the standard scheme) at the next layer after ascent, while taking entrainment

of environmental air into consideration. The convective process is then initiated. The parcel

continues to rise, entraining environmental air and detraining cloudy air, until it is no longer

buoyant after being lifted from layer k to the next model layer k+1. It is then assumed that a

proportion of the plumes represented by the parcel have reached zero buoyancy in layer k and

have detrained there, the proportion being just sufficient to allow the parcel to be buoyant by an

amount b (defined above) in layer k+1 . Hence allowance is made for the varying characteristics

of clouds within the assumed ensemble while still retaining a simple ‘bulk’ cloud model approach.

The parcel ascent continues until the zero buoyancy level of an undilute parcel from the starting

layer of convection is reached or the convective mass flux falls below a minimum value. In each

case the parcel is totally detrained into the cloud environment.

4

(2) CLOUD MODEL

(a) Definition of the bulk cloud model

Although a ‘bulk’ cloud model is used in the scheme this is derived by summation over an

ensemble of convective clouds with differing characteristics. For a cloud I within the ensemble the

equations governing cloud mass flux (M, Pa/s), potential temperature (�, K) mixing ratio (q,kg/kg)

and cloud liquid water (l, kg/kg) as the parcel rises are

(1)

(2)

(3)

(4)

where M = cloud mass flux p � =

� in cloudy air P

�

= �

in environmental air E

�= �

on mixing detrainment N

�= �

on forced detrainment R

E = entrainment rate

N = mixing detrainment rate

D = forced detrainment rate

Q = conversion of water vapour to liquid water and ice

PR = liquid water and ice precipitated �

= �

for cloud I I� = (p/100000) and � = R/c (p is in Pa)

�p

R = gas constant for dry air

c is the specific heat of dry air P

and L’ is the latent heat of condensation or condensation plus fusion

(if freezing occurs)

Entrainment and detrainment rates are given by,

5

(5)

following Turner(1963), where � , µ, and � are fractional mass entrainment and detrainment

coefficients.

These equations are similar to those presented by Yanai et al (1973) (equations 27 to 30) although

differences in the treatment of detrainment must be noted. In the above equations two detrainment

processes have been included. The first, mixing detrainment (at a rate N ) represents the I

detrainment of cloud air through turbulent mixing at the edge of the cloud and occurs even when

the cloud is positively buoyant. Forced detrainment (at a rate D ) occurs at cloud top when theI

cloud reaches zero buoyancy. It is therefore zero until this level.

Following Yanai et al (1973) and integrating over all cloud types leads to

equations for the ‘bulk’ cloud model;

(6)

(7)

(8)

(9)

where

and

6

� ,

� and

� are defined as mass weighted averages over the entire ensemble of cloud, the airP N R

undergoing mixing detrainment and that undergoing forced detrainment.

(b) Discretised equations

Figure 1 illustrates how these equations are discretised in the vertical for layer k (k decreasing

upward). Mixing and any forced detrainment occur at level k while environmental air is entrained at

levels and . To simplify the cloud model it is assumed that air undergoing

mixing detrainment has the characteristics of the ensemble mean cloud at the same level, thus

avoiding specification of cloud edge properties. Condensational heating is added at level k+1. The

discretised forms of equations (6), (7) (8) and (9), after some manipulation, are

MASS

(10)

POTENTIAL TEMPERATURE

(11

)

MOISTURE

(12

)

CLOUD WATER

7

(13)

where �

is �

in layer k, K

�

is �

in layer k+1, K+1

�

is �

on forced detrainment in layer k, RK

�

is parcel �

in layer k PK

COND is the water vapour sink due to condensation at level k K

PREP is the amount of liquid water precipitated from the parcel at level k K

�

= (p /100000) and � = R/C (p is in Pa), K K p

�

,

,

and , L being the latent heat of condensation

or , L being the latent heat of fusion F

The choice of cloud water freezing point as 263.15K allows for the presence of supercooled water

drops within cloud.

It should also be noted that the definitions of the various Dp’s above is in the opposite sense to

that of UM Documentation Paper No. 10 and removes the necessity of minus signs on the lhs of

equations (1) to (4) and (6) to (9) in the discretised equations.

These equations can be rearranged to the parcel values in layer k+1 after lifting from layer k;

(10a)

(11a)

(12a)

8

(13a)

where

These equations are implicit, are dependent upon . The method

by which they are solved is;

(i) A dry ascent, taking entrainment into account but not forced detrainment is carried out from

layer k to k+1 to determine

(ii) If then the parcel is saturated in layer k+1 and

is calculated by adding the condensation term.

(iii) if then forced detrainment is initiated, � beingK

calculated and the final potential temperature of the parcel being derived.

(c) Entrainment and detrainment

Several studies have been undertaken to determine the entrainment rate associated with

convective plumes. Simpson and Wiggert (1969) and Simpson (1971) suggest a fractional

entrainment coefficient which depends upon cloud radius R (for a height co-ordinate model):

Transforming to pressure co-ordinates, .Assuming g 10ms and

. For shallow clouds with while for deep

clouds with . Hence some variation of entrainment coefficient

with height seems desirable. Near the surface both shallow and deep clouds will exist together,

while if a plume reaches the upper troposphere only deep clouds remain. In the scheme the

choice of

9

is made. A (defined below) is usually set to 1.5 and so near the surface p p and � 4.5x10 E *-5

reflecting the presence of both shallow and deep convection.At p=20000Pa,

� 1x10 (assuming p � 100000Pa) representing entrainment into deep convective towers. This gives-5*

a variation of entrainment rate with height similar to that diagnosed by Yanai et al(1973) (figure 14

of their paper), as illustrated in figure 2 which compares their diagnosed rate to that of the present

formulation calculated using the convective mass flux from figure 13 of Yanai et al (1973).

In the discretised equations the two fractional entrainment coefficients are defined at k+1/4 and

k+3/4 as

(14)

where A =1 if k=1 (lowest layer) E

and A =1.5 if k>1 E

The mixing detrainment coefficient for cloudy air is defined by

(15)

The choice of A above implies that µ is zero in the lowest model layer. E K

(3) INITIAL ‘DRY’ ASCENT

Starting from the lowest model layer each layer is tested in turn until one is found from which

convection can be initiated. However before carrying out a detailed parcel ascent to determine

whether convection might occur from a layer k, a simple stability test is carried out to determine if

the profile is too stable for any convection to develop from that model layer.

An approximate parcel potential temperature in layer k+1 is calculated based upon the

assumptions that no entrainment occurs during lifting and that the mixing ratio of the parcel if

saturated in layer k+1 is equal to the saturation mixing at the layer’s potential temperature;

(16a)

If, (16b)

where DETHST = 1.5K in the standard scheme, then it is deemed that convection may be possible

from layer k and a more accurate calculation of the parcel ascent is carried out as described

below.

For convection starting from layer k the initial parcel potential temperature and mixing ratio are

given by,

10

(17)

where s = 0.2K.

The parcel is lifted from layer k to k+1 initially with only with entrainment processes being taken

into account. The parcel potential temperature, mixing ratio are given by,

(11b)

(12b)

Equations (11b) and (12b) are also used to calculate the first part of the ascent from layer k to k+1

if convection already exists in layer k-1. In this case are replaced by the

final parcel potential temperature and mixing ratio from the ascent from layer k-1 to k.

(4) CONDENSATION

If during the initial dry ascent the lifting condensation level is exceeded then the parcel is saturated

in layer k+1 and the potential temperature must be adjusted through the addition of the term COND

(in equations (11) and (12)). COND is given by, K+1

(18)

where � = 1 if parcel in layer k+1 is saturated K+1

ie

or =0 if the parcel is unsaturated in layer k+1

Equation (18) implies that the calculation of is implicit. An approximate solution is

found by the truncated Taylor expansion for ;

(19)

is obtained from the Clausius-Clapeyron equation;

11

(20)

can be obtained by a rearrangement of equation (11a) using equations (11b), (18),

(19) and (20) again assuming the forced detrainment rate is zero;

(21)

is taken to be the saturation mixing ratio at .

(5) CALCULATION OF INITIAL MASS FLUX

The mass flux of the plume in the initial convective layer (M ) is simply related to the stability of theI

lowest convecting layers by the empirical formula;

(22)

where c is 3.33X10 -4

d is 0.0

p is in Pa

p* is the surface pressure (in Pa)

M is in Pas I -1

and is the virtual potential temperature of the parcel in layer

k+1 after entrainment and latent heating effects have been considered.

Thus the initial mass flux is proportional to the excess buoyancy of the parcel starting from layer k

in layer k+1.

With the closure formulated above parcel buoyancy is the only criteria which determines the

development of convection. Hence convective clouds can occur without the presence of

large-scale convergence (unlike the Kuo and Arakawa-Schubert schemes), although would quickly

cease without the convergence compensating the warming and drying due to convection above the

boundary layer.

The constant ‘c’ determines the convective mass flux per unit increase in parcel buoyancy with

height across the initial convecting layer. Hence the larger the value of ‘c’ the more convective

mass flux is generated for a given atmospheric stability. The value of ‘c’ is chosen to give a

12

realistic atmospheric thermodynamic structure and surface rainfall rate and also to ensure that

there was not an excessive growth in conditional instability within a column of the atmosphere,

which might lead to the risk of computational instability.

(6) FORCED DETRAINMENT

With the above formulation of entrainment and mixing detrainment described in section 1.2(c) the

convective massflux increases with height as expected for a buoyant parcel. However this is

restricted by the process of forced detrainment which represents the terminal outflow from those

members of the ensemble which have reached zero buoyancy.

If on ascending from layer k to k+1 (after entrainment and moist processes have been taken into

account) the excess buoyancy of the parcel is less than a minimum value (b) ie.,

(23)

then forced detrainment is initiated, a portion of the parcel being detrained in layer k representing

terminal detrainment from the less buoyant members of the ensemble.

The magnitude of b (0.2K in the standard scheme) determines at what level in any ascent forced

detrainment first occurs. Weaker members of the ensemble may have reached zero buoyancy well

below this level and hence their level of terminal detrainment will be elevated. This is offset by the

processes of mixing detrainment which occurs even when the parcel is positively buoyant.

On forced detrainment the virtual potential temperature of the detraining air in layer k is set equal

to that of the cloud environment;

(24)

can be obtained from truncated Taylor expansion,

(25)

However this leads to a quadratic in �

. Replacing in the first termR

of equation (24) makes the equation linear in �

. This approximation incurs a fractional error inR

the moisture term of the virtual potential temperature of . Using the alternative

approximation of incurs a larger error of .

13

For a saturated parcel, solving for �

from equation (24) and (25), R

(26a)

For an unsaturated parcel equation (24) is linear in , assuming that is equal to mixing

ratio of the unsaturated parcel mean mixing ratio and so on rearrangement,

(26b)

To determine the revised parcel temperature in layer k+1 the fraction of the parcel mass detrained

in layer k is needed. With a bulk cloud model this is not known and an alternative approach must

be taken. It is assumed that the fraction of the parcel detrained in layer k is that which allows the

parcel to be buoyant by an amount b in layer k+1;

(27)

This is solved in the same manner as equation (24) to give for a parcel saturated in layer k+1,

(28a)

while for an unsaturated parcel,

(28b)

Again the choice of 0.2K for b implied that the ‘bulk’ parcel is just buoyancy in layer k+1 after

forced detrainment. Yanai et al (1973) suggest a larger value is appropriate; using data from the

west Pacific they diagnosed a mean ensemble temperature excess above the cloud environment of

2-3K throughout much of the convective layer. However recent studies using explicit convective

cloud models to simulate cloud ensembles (Gregory (1986), Tao et al (1987)) indicate an

ensemble mean temperature excess of only 0.5-1K in the cloud layer.

If in the original ascent the parcel was only just saturated and near the environmental potential

temperature it is possible that,

(29)

14

This implies that there is insufficient condensation in lifting the parcel from layer k to k+1 for it to be

saturated after forced detrainment. In this instance is recalculated as for an unsaturated

parcel (eqn(28b))

The temperature of the parcel in layer k+1 after forced detrainment may be lower than that

calculated neglecting the forced detrainment process. In this instance the detrainment calculation is

abandoned and the state of the parcel returned to that at the end of the moist ascent (section(1.4),

eqn(21)).

The detrainment coefficient � is diagnosed using equations (10), (11) and (12) together withK

values of from equations (26) and (28). In doing so adjustments have to be

made for additional condensation which occurs in saturated air which is being detrained from the

parcel in layer k+1, as explained below.

In general is not equal to and so for a saturated parcel in layer k will not be equal

to (which are the saturation mixing ratios at their respective temperatures). Hence in

lifting a parcel from the initial convecting layer to layer k either more or less water must have

condensed out of the air undergoing forced detrainment than in the mean parcel. In reality this

increase or decrease in condensation will have occurred gradually during the ascent to layer k but

in the scheme it is assumed to occur in layer k. The change in condensation is given by;

(30)

If the parcel values in layer k are adjusted to account for this extra latent heating the parcel ascent

from layer k to k+1 would have to be repeated. Adjustment is therefore made to the parcel

characteristics in layer k+1 which are used to calculate the forced detrainment rate.

Equation (11a) is modified by the addition of an extra latent heating term given by eqn (30), the

final equation for used in the calculating the forced detrainment rate being,

(11c)

where is given by

15

(18a)

It should be noted that equation (11c) is not a prognostic equation for the parcel temperature in

layer k+1 but a diagnostic equation for the forced detrainment rate. On rearrangement the forced

detrainment rate for layer k is given by;

(31a)

where

(31b)

Once the detrainment rate has been calculated the moisture budget of parcels saturated in layer

k+1 is checked. A new value of is calculated, adjusting for the change in condensation

in layer k due to forced detrainment;

(12c)

If is found to be less than the parcel mixing ratio in layer k+1 after forced

detrainment then the calculation of is repeated but assuming the

parcel is unsaturated in layer k+1.

The above formulation of forced detrainment should require the forced detrainment rate to be

bounded by unity and zero. However because of the approximations used in the calculation and

the numerical discretisation, it may be possible for the forced detrainment rate to exceed these

boundaries. If the forced detrainment rate is found to be less than zero or greater than unity,

again the forced detrainment calculation is abandoned and parcel values in layer k+1 restored to

those calculated at the end of the moist ascent (section(1.4), eqn(21)).

(7) TERMINATION OF CONVECTION

If the distribution of parcel temperatures in the ensemble was unbounded forced detrainment

could increase the ‘bulk’ parcel temperature indefinitely, detraining progressively smaller amounts

of mass with ever increasing temperatures. However in reality the maximum parcel temperature of

any member within the ensemble must be that of an undilute parcel from the initial convective

starting level. Thus the parcel is totally detrained once the level of zero buoyancy of such an

undilute parcel is reached. The potential temperature of an undilute parcel in layer k+1 is

16

approximated by,

(32)

where is the value of í in the initial convecting layer,

and is an approximate value for the saturation mixing ratio at 263K, the temperature at

which the condensate freezes

It is also assumed that if the convective mass flux falls below a minimum value defined by,

(33)

where E is 1K in the standard scheme, and c is the constant used to determine the initialMIN

parcel massflux and is set to 3.33x10 (see section (1.3)), then the parcel is destroyed by

entrainment.

When convection terminates, the fraction of the parcel still detrains in layer k

with neutral buoyancy, the remaining fraction detraining in layer k+1 with

parcel values. (In the original scheme total detrainment of the parcel occurred in layer k with

parcel values. This did not allow sufficient mixing between the boundary layer and the free

atmosphere in regions of extensive shallow cumulus (e.g. the trade wind regions) producing too

strong and too dry an inversion above a nearly saturated boundary layer. Slingo et al (1989)

describes the impact of these changes upon the 11-layer model in detail).

(8) PRECIPITATION PROCESSES

The development of precipitation in shower clouds occurs mainly by the coalescence of smaller

drops of water with larger ones. As discussed by Ludlam (1980, chapter 8) the rate at which this

occurs depends upon the distribution of the aerosol size spectrum, so that clouds which form in air

masses with greater numbers of large nuclei (r>20µm) present precipitate earlier. This is reflected

in the differing depths that clouds have to attain in maritime and continental air masses for

precipitation to be initiated. This depth is greater for nucleus-rich continental air (3 to 5 km) than

for maritime air (1 to 2 km) which has fewer but typically larger condensation nuclei (table 8.3 in

Ludlam (1980, chapter 8)). Although the glaciation of a cloud is not necessary for the formation of

precipitation, clouds whose tops exceed the -10 C level by 1 km will produce precipitation by the

growth of hail from the freezing of supercooled water droplets (Ludlam 1980, chapter 8).

17

The aerosol distribution of air is not available in the model but these observations are reflected in

the convection scheme by suppressing the precipitation of condensed water until the cloud has

exceeded a critical depth, D (base to top) and the amount of cloud condensate within the CRIT

parcel (l exceeds a minimum value l . Until these criteria are met no precipitation occurs andK MINP

the condensed water is stored within the parcel as the ascent continues. In the standard scheme

l is set to 10 kg/kg or the local saturation specific humidity where it is less, while D isMIN CRIT-3

defined by,

D = (34)CRIT

These crudely take into account variations of atmospheric aerosol over land and sea and also the

observation that precipitation forms in clouds soon after glaciation.

The cloud depth to layer k (in m) is calculated by the integral of the hydrostatic equation in the

vertical;

(35)

where k is the cloud base layer, and � � is the difference in the exner ratio across layer K b K

If these criteria are met, the amount of precipitation produced, in taking the ensemble from level k

to k+1 is,

(36)

l of the condensate is retained by the parcel in the process. Total precipitation falling fromMIN

cloud base is obtained by summing P between cloud base and cloud top. K+1

(9) CONVECTIVE CLOUD BASE, TOP AND AMOUNT

These quantities are calculated by the convection scheme for use in the radiation calculation of the

model. The cloud base is taken to be the lower boundary of the first model layer at which

saturation occurs, whereas cloud top is the upper boundary of the last buoyant layer.

The fractional convective cloud amount is given by the empirical formula,

C = 0.7873 + 0.06ln(TCW) (%) (37) CONV

where TCW = , k = cloud base layer, k = cloud top layer, the mass of b t

liquid water condensed per unit area between cloud base and

18

top (kg/m ), 2

l is the condensed water content of the parcel (liquid or ice) beforeKP

precipitation

and � t is the model timestep

If TCW < 2.002x10 then C is set to zero. C is also never allowed to exceed unity. -6CONV CONV

(10) IMPACT ON ENVIRONMENT

As with all mass flux schemes the plume interacts with its environment through en/detrainment of

heat, moisture and cloud liquid water and subsidence induced in the cloud environment which

compensates for the parcel’s upward mass flux (see Gregory and Miller (1989) for the theoretical

derivation of the cumulus heating and moistening rates due to these processes). Considering

fluxes of �, q and l between cloud and environment in layer k and vertically out of and into layer k

(fig 1), the rate of change of the environmental potential temperature in K/s, and specific humidity

in kg/kg/s are given by,

(38

a)

(3

8b)

The first term in these expressions represents compensating subsidence, the second and third

forced and mixing detrainment respectively. The last term is the effect of the detrainment of cloud

condensate. Changes of phase and evaporation of precipitation also contribute to the

environmental temperature and moisture changes and are considered in the following section.

In the layer that convection is initiated the heat required to raise the initial parcel temperature from�

to �

+ s is removed from the environment; E E

(39)

When convection terminates split final detrainemnt occurs and convection also increments the

potential temperature and mixing ratio in layer k+1 due to the detrainment of cloud air;

(40)

(40b)

19

(11) CHANGES OF PHASE AND EVAPORATION OF FALLING PRECIPITATION

Precipitation from the scheme may be either water (rain) or ice (snow) depending upon the

temperature of the layer through which the precipitation falls, changes of phase occurring when the

freezing/melting level (273.15K) is crossed. This is different from the criteria used to determine the

phase of condensate within cloud (section 1.2(b)) and so before precipitation falls to the surface

from a layer k it is possible for a change of phase to occur according to the temperature of the

cloud environment in layer k. This increments the potential temperature of layer from which the

precipitation starts to fall;

(41)

where P is the precipitation starting to fall from layer k. K

Further changes of phase occur falling precipitation crosses the freezing/melting level (273.15K),

and heat released (by freezing) or lost (by melting) is added to the model potential temperature of

the layer physically below the freezing level. The increment is given by,

(42)

where J is the precipitation falling into layer k from k+1. K+1

The sign in equations (34) and (35) depends on whether freezing or melting occurs.

Falling snow is not evaporated but falling rain is allowed to evaporate below cloud base; the rate is

proportional to the sub-saturation of the environment and is constructed by integrating the fractional

evaporation of precipitation over a layer of depth dz ie,

(43)

where � is the density of air, � is a constant set to 10- s in the standard scheme (based upon3 -1

Kessler (1969)) and Q is the local rainfall rate estimated by assuming precipitation covers 10% of a

grid box, a typical value suggested by an analysis of GATE phase 3 ship and radar rainfall data.

Hence Q=10J where J is the grid box mean rainfall at cloud base. B B

If the air is not saturated the precipitated water J reaching level k from level k+1 is given by, K

(44)

where J is in Kg/m /s, q in kg/kg and � p is in Pa. K K2

Mixing ratio and potential temperature increments due to evaporation in layer k are;

(45a)

20

(45b)

(12) ENERGY CONSERVATION AND UPDATING OF MODEL VARIABLES

Once all the increments have been calculated a check is made to ensure enthalpy is conserved,

(46)

where L is the latent heat of fusion, F

q is the ice content, F

and � is the increment due to changes in phase only. S

If T ≤ 273.15K ie the temperature of the lowest model layer is less than the freezing point of1

water, all the water condensed has turned to ice and so

and equation (46) becomes

(46a)

Otherwise when T > 273.15K any ice formed by freezing of condensed water will have melted (in1

the case of falling snow) or sublimated (in the case of detrained cloud ice) and

giving

(46b)

In general due to the inaccuracy in calculating the temperature along a moist adiabat and also the

discretisation of the cloud model in the vertical the rhs of equations (46a) and (46b) are non zero;

(47)

where L’ = L if T > 273K, 1

and = (L+L ) if T ≤ 273K. F

The rhs of equation (47) is reduced to zero by adjusting those temperature increments which are of

the same sign as the residual.

21

If RES > 0 then

(48a)

where (49a)

and if RES < 0 then

(48b)

where

(49b)

In equations (48) and (49) ( � T/ � t) refers to those increments which are positive while ( � T/ � t) + -

to those which are negative. If the summation of the increments which are the same sign as the

residual is zero then increments of the opposite sign are adjusted. If both summations are zero

then no adjustment is made.

Once the potential temperature increments have been corrected to conserve enthalpy the model

potential temperature and mixing ratio fields are updated;

(50a)

(50b)

where � t is the model timestep

22

REFERENCES FOR PART 1

Gregory, D., 1986 : A numerical study of the parametrisation of deep tropical convection., Ph.d

thesis, University of London, Gower Street, London.

Gregory, D., and Miller, M.J., 1989 : A numerical study of the parametrisation of deep tropical

convection., Quart. J. Roy. Meteor. Soc., 115, pp1209-1242

Gregory, D., and Rowntree, P.R.R., 1990 : A mass flux convection scheme with representation of

cloud ensemble characteristics and stability dependent closure., Submitted to Mon. Wea. Rev.

Kessler, E., 1969 : On the distribution and continuity of water substance in atmospheric

circulations., Met. Mon., (Amer. Met. Soc.), 10, N .32.

Ludlam, F.H., 1980 : Clouds and Storms., Pennsylvania State Univ. Press, University Park,

Pennsylvania, USA, 407pp plus plates

Lyne, W.H., and Rowntree, P.R., 1976 : Development of a convective parametrization using GATE

data., Met O 20 Technical Note II/70, Meteorological Office, Bracknell, UK.

Simpson, J., 1971 : On cumulus entrainment and one dimensional models., J. Atmos. Sci., 28,

pp449-455

Simpson, J., and Wiggert, V., 1969 : Models of precipitating cumulus towers., Mon. Wea. Rev., 97.

pp471-489

Slingo, A., 1985 : Handbook of the Meteorological Office 11-layer atmospheric general circulation

model. Volume 1 : Model description., Dynamical Climatology Technical Note No. 29,

Meteorological Office, Bracknell, UK.

Tao, W-K., Simpson, J., and Soong, S-T., 1987 : Statistical properties of a cloud ensemble : A

numerical study., J. Atmos. Sci., 44, pp3175-3187

Turner, J.S., 1963 : The motion of buoyant elements in turbulent surroundings., J. Fluid Mech.,

16,1-16.

Yanai, M., Esbensen, S., and Chu, J.-H., 1973 : Determination of bulk properties of tropical cloud

clusters from large-scale heat and moisture budgets., J .Atmos. Sci., 30, pp611-627

23

LIST OF FIGURES IN PART 1

Fig 1 Discretisation of the convection scheme in the vertical

Fig 2 Comparison of mass entrainment rates diagnosed by Yanai et al (1973)

(-----) with the present formulation (- - -) calculated using the convective

massflux from Yanai et al (1973)

24

25

Part 2 - description of Downdraught

Contents

2.1. Introduction

2.2. Modifications to the updraught

2.3. Diagnosis of and initialisation of the downdraught

2.4. Descent of parcel(a) Dry descent(b) Precipitation within downdraught(c) Change of phase(d) Evaporation of precipitation

2.5. Termination of downdraught

2.6. Effect of downdraught upon environment

2.7. Treatment of non-downdraught precipitation

2.8. Energy conservation

26

2.1. Introduction

Deep convective clouds contain both upward and downward motion on the cloud scale. Thedownward motion is driven by the evaporation of falling precipitation and also by thedownward drag of the water. These features play an important role in the energetics of thecloud system and especially in the interaction of deep convection with the sub-cloud layer andhence surface processes.

This paper describes the parametrization of deep convective downdraughts used in the UnifiedModel. The downdraught is based upon an inverted entraining plume whose negativebuoyancy is maintained by the evaporation and sublimation of falling precipitation andchanges of phase. Slight modifications to the updraught part of the scheme (from that at 1A)have been made in formulating this extended scheme and these are also described.

The scheme described here is version 3A of the code in the UM library. The original versionof the scheme is code 2A while 2B contains a modified method to initiate the updraught (seeGregory (1995) - UM Documentation paper 27 part 1). Differences between the downdraughtscheme in version 3A of the convection scheme and earlier versions are noted below.

2.2. Modifications to the updraught

The following modifications have been made to the updraught part of the scheme inimplementing the downdraught part of the scheme.

(a) Condensate now changes phase at 0oC rather than -10oC

(b) The coefficient used to calculate the initial convective (updraught) mass flux (c) isincreased to 5.17x10-4 on the basis of single column model experiments with GATE data.

(c) The evaporation/sublimation of precipitation not falling through the downdraught isupdated to be consistent with the treatment of evaporation/sublimation of precipitation withinthe downdraught.

(d) Unlike the original scheme the downdraught formulation allows falling snow (rain) toexist when the environment temperature of the cloud is above (below) the freezing point ofwater . This necessitates a change to the energy conservation procedure (section 2.8) as thesurface layer temperature cannot be used to determine the phase of the falling precipitation.

27

2.3. Diagnosis of and initialisation of the downdraught.

Deep convective downdraughts are allowed to exist when the updraught satisfies the followingconditions;

(i) Cloud top pressure is at least 150mb above the surface.(ii) The updraught is saturated in the layer in which it terminates.(iii) The depth of the cloud layer exceeds 150mb.

Also, even if these conditions are met, downdraughts are not allowed to form within 150mbof the surface.

Version 2A and 2B of the "downdraught" scheme used similar criteria to those above in orderto determine where downdraughts would initiate but based upon model levels not pressure.In these earlier versions downdraughts would initiate if;

(i) Cloud top exceeds model level 4.(ii) The updraught is saturated in the layer in which it terminates.(iii) The depth of the cloud exceeds two model two model layers.

Downdraughts were not allowed to initiate within 4 model layers of the surface.

Once these conditions are met the initial downdraught mass flux (Pas-1) is calculated from the

(1)M DDinit αM UD

ref

updraught mass flux at a reference level;

where α is chosen to be 0.05 on comparison with the scheme with explicit cloud modelresults using a single column version of the Unified Model. The reference level is chosento be 3/4 the way up the depth of the cloud and this mass flux is used regardless of the levelthe from which the downdraughts initiated.

Starting from the lowest layer above cloud top (the highest layer into which cloud detrains)and working downward to within 150mb of the surface, ‘each layer is tested to see if adowndraught may be initiated. The downdraught air is assumed to be an equal mixture ofcloudy updraught and unsaturated environmental air. The potential temperature (K) andmixing ratio (kgkg-1)of this unsaturated parcel are;

(2)θDDi unsat k

12

(θUDk θE

k)

28

(3)q DDi unsat k

12

(q UDk q E

k )

The unsaturated parcel is then assumed to be brought to saturation through theevaporation/sublimation of precipitation (although no adjustment is made to the amount ofprecipitation produced in the updraught). The saturation point is calculated by a Taylorexpansion;

where L’ = LC or LC+LF depending upon the phase of the precipitation.

(4)θinit sat k DD θDDinit unsat k

LcpΠk

(qs(θDDinit sat k) q DD

init unsat k)

Now,

where θFG is a first guess temperature and,

(5)

qs(θDDinit sat k) qs(θFG (θDD

init sat k θFG))

qs(θFG (θDDinit sat k θFG)

⎛⎜⎜⎝

⎞⎟⎟⎠

∂qs

∂θ FG

)

(6)⎛⎜⎜⎝

⎞⎟⎟⎠

∂qs

∂θ FG

⎛⎜⎜⎝

⎞⎟⎟⎠

LRv

qs(θFG)

Πkθ2FG

the Clausius-Clayperon equation.

Hence after rearrangement,

Two iterations of this calculation are carried out to achieve sufficient accuracy. On the first

(7)θDDinit sat k

⎛⎜⎜⎜⎝

⎞⎟⎟⎟⎠

θDDinit unsat k (L /cpΠk)(qs(θFG) q DD

init θFG

⎛⎜⎜⎝

⎞⎟⎟⎠

∂qs

∂θ FG

⎛⎜⎜⎜⎝

⎞⎟⎟⎟⎠

1 (L /cpΠk)⎛⎜⎜⎝

⎞⎟⎟⎠

∂qs

∂θ FG

iteration θkE is used as the first guess temperature. On the second iteration θinit sat k

DD from

29

the first iteration is used.

The buoyancy of the saturated parcel is then estimated taking into account virtual effects (butnot water loading);

(8)B DDk θDD

init sat k(1 Cvirqs(θDDinit sat k)) θE

k(1 CvirqE

k )

where Cvir = (1/0.62198)-1.

If BDDk < 0 then a downdraught is initiated from layer k with an initial potential temperature

given by equation (7) and the mixing ratio set to the saturation mixing ratio at thistemperature.

30

2.4. Parcel descent

If an initially saturated parcel is displaced downwards it will gradually become unsaturateddue to adiadatic warming. However, falling precipitation in the downdraught will evaporatemoving the parcel thermodynamic characteristics towards saturation. In reality theseprocesses occur simultaneously but in the scheme they are modelled in the following orderas the parcel descend from layer k to k-1;

i) Dry descent,ii) Change of phase of precipitation according to the saturation temperature in layer k-1,iii) Evaporation/sublimation of falling precipitation into the sub-saturated downdraught.

a) Dry descent

The parcel undergoes dry descent from layer k to k-1, undergoing mixing entrainment anddetrainment. The mass continuity and thermodynamic equations for the dry part of thedescent are descritized in the vertical as illustrated by figure 1 and have the form,

where ∆pk-1/2 = pk-1 - pk, ∆pk-1/4 = pk-1/2 - pk and ∆pk-3/4 = pk-1 - pk-1/2

(9)M DDk 1 M DD

k (1 ηDDk ∆pk 1/2)(1 εDD

k 1/4∆pk 1/4)(1 εDDk 3/4∆pk 3/4)

(10)M DD

k 1 θDD DRYk 1 M DD

k θDDk ηDD

k ∆pk 1/2MDD

k θDDk

εk 1/4∆pk 1/4MDD

k (1 ηDDk ∆pk 1/2)θ

Ek

εk 3/4∆pk 3/4(1 ηDDk ∆pk 1/2)(1 εk 1/4∆pk 1/4)M

DDk θE

k 1

The equation for moisture follows a similar form to eqn (10).

The basic entrainment rate for the downdraught is assumed to be constant with height;

where p* is the surface pressure (Pa) and AE=1.5, CDD=3.

(11)εk 1/4 εk 3/4

AECDD

p

These entrainment rates are enhanced at the freezing level, justified by the observation thatmost downdraughts are observed to form near to this level in the atmosphere where aminimum in equivalent potential temperature usually exists, such air is the most unstable toa downward displacement. The factor by which entrainment rates are enhanced is given by;

31

(12)IFAC 1.8x10 6M DD

init

CDDp

This factor is limited to a maximum value of 6 on the basis of single column model tests.

The formulation of entrainment is rather empirical but does give reasonable results againstGATE data in single column model studies and also against explicit cloud model results (seeGregory and Bett, 1991).

The detrainment formulation is related to the entrainment rates but varies with height. Above

(13)ηk

1CDD

(εk 1/4∆pk 1/4 εk 3/4∆pk 3/4)

∆pk 1/2

(1 1AE

)

the freezing level,

while at the freezing level and below this,

(14)ηk

(εk 1/4∆pk 1/4 εk 3/4∆pk 3/4)

∆pk 1/2

(1 1AE

)

Again this was chosen on the basis of comparison of downdraught cooling profiles predictedby the scheme with those diagnosed from explicit cloud models.

It should be noted that if the downdraught penetrates to within 100mb of the surface then amodified entrainment/detrainment strategy is used. Entrainment rates are set to zero whilethe proportion of downdraught air detrained into each layer is proportional to the layerthickness;

where pk+1/2 is the pressure of the top boundary of layer k.

(15)ηk

∆pk

(p pk 1

2)

Note that if the freezing level lies within the detrainment layer then entrainment rates are notenhanced by the factor IFAC.

32

b) Precipitation within the downdraught

Adiabatic warming due to dry descent would soon force the downdraught to be positivelybuoyant and so terminate the descent. Deep convective downdraughts are maintained atnegative buoyancies through the evaporation/sublimation of falling precipitation. Waterloading effects are also important but are not considered in the present scheme.

A proportion of the precipitation created in the updraught is assumed to fall through thedowndraught, undergoing evaporation and sublimation as it does so. The remainder of theprecipitation is assumed to fall through cloudy air, only undergoing evaporation/sublimationbelow cloud, although some of this precipitation is allowed to enter the downdraught as itdescends.

The amount of precipitation transferred from the updraught to the downdraught at any levelk is calculated on the basis of maintaining continuity of precipitation mixing ratios betweenupdraught and downdraught. It would be unreasonable for the mixing ratio of precipitationwithin the downdraught to be larger than that within the updraught where the precipitationoriginated. Hence it is assumed that,

where lp is the mixing ratio of precipitation and [ ]Input refers to the value on entry into the

(16)[l pk ]DD

Input [l pk ]UD

downdraught.

After rearrangement this can be written in terms of the precipitation produced in theupdraught;

where PkUD is the precipitation produced in the updraught due to ascent from layer k-1 to

(17)[l pk ]DD

Inputg

M UDk

P UDk

layer k (see UM Documentation paper 27 part 1 for further discussion of precipitationgeneration within the updraught).

Hence after injection of precipitation into the downdraught from the updraught in layer k thedowndraught precipitation mixing ratio is,

where Fk is the fraction of precipitation which is transferred from the updraught into the

(18)[l PK ]DD2 [l P

K ]DD g

M UDk

P UDk [l P

K ]DDFk

M DDk

P UDk

downdraught and [lPk]DD is the downdraught precipitation mixing ratio falling into layer k

from above.

33

If no precipitation is produced within the updraught at the level under consideration thencontinuity in the vertical is maintained with the updraught precipitation mixing ratio at thelast layer in which precipitation was generated.

Continuity of downdraught precipitation mixing ratio as the parcel descends is also assumedand so the mass of precipitation within the downdraught is constrained by,

To achieve this non-downdraught precipitation, i.e. that which is not assumed to have been

(19)M DDk 1 [l P

k 1]DD M DD

k 1 [l Pk ]DD2

transferred from the updraught to the downdraught in the layer in which it was formed, isused;

where the summation term on the rhs is the precipitation falling outside of the downdraught

(20)M DDk 1 [l P

k ]DD2 M DDk [l P

k ]DD2 FekΣkctj k 1Pj(1 Fj)(1 Fej)

and Fek is the fraction of that precipitation which is required to maintain continuity ofdowndraught precipitation mixing ratio as the parcel descends between levels k and k-1.

On rearrangement of equation (20) Fek may be estimated;

Fek may be positive or negative depending upon whether the downdraught mass flux is

(21)Fek

(M DDk 1 M DD

k )[l Pk ]DD2

Σkctj k 1Pj(1 Fj)(1 Fej)

growing or shrinking with height. Its absolute value is limited to be less than unity.

If no downdraught exists then all precipitation is assumed to fall through cloudy air abovecloud base. If the downdraught terminates above the surface layer then all the remainingdowndraught precipitation is added to that which is not falling through the downdraught.

c) Change of phase.

As precipitation falls through the downdraught it undergoes changes of phase according tothe state of the precipitation and the temperature of the downdraught. Because the descentfrom the layer k to k-1 is discretised several downdraught temperatures are calculated duringthe procedure. The temperature used to determine the change of phase is the saturationtemperature of the parcel due to evaporation/sublimation after the dry descent. This iscalculated after the manner in section (2.3) (eqn 7) but using the model environment potentialtemperature in layer k-1 (θE

k-1) as the first guess temperature.

Melting of snow occurs if the saturated temperature in layer k-1 is greater than OoC. Thetemperature of the unsaturated parcel after melting is not allowed to cool below the freezing

34

point and the amount of precipitation melted is given by,

(22)∆Sk 1 min(cp

Lf

(T DD DRYk 1 Tmelt)

M DDk 1

g,S DD

k 1 )

with Tmelt=273.15K.

The new temperature of the downdraught after melting is

and the precipitation is adjusted accordingly,

(23)θDD DRY 2k 1 θDD DRY

k 1 gLF

cp

∆Sk 1

Πk 1MDD

k 1

Rain is assumed to freeze when the saturated temperature in layer k-1 is less than OoC on the

(24)R DDk 1 R DD

k 1 ∆Sk 1

S DDk 1 S DD

k 1 ∆Sk 1

condition that θDD dryk-1 is also less than 0oC. The temperature of the unsaturated parcel after

freezing is not allowed to exceed 0oC, the amount of precipitation undergoing freezing beinggiven by,

The new temperature of the downdraught after freezing is,

(25)∆Rk 1 min(cp

Lf

(Tmelt T DD DRYk 1 )

M DDk 1

g,R DD

k 1 )

and the adjusted precipitation,

(26)θDD DRY 2k 1 θDD DRY

k 1 gLF

cp

∆Rk 1

Πk 1MDD

k 1

Because the melting and freezing processes are limited when the parcel temperature crosses

(27)R DDk 1 R DD

k 1 ∆Rk 1

S DDk 1 S DD

k 1 ∆Rk 1

35

the freezing point it is possible for rain and snow to exist together within the downdraught.This allows the possibility of snow reaching the surface when the lowest model leveltemperature is above 0oC and for rain to exist below the same temperature.

d) Evaporation of precipitation.

Before the evaporation calculation a revised saturation temperature for the downdraught inlayer k-1 is calculated again using θE

k-1 as the first guess temperature. This accounts forchanges induced by the melting and freezing process.

It is possible that at the end of any melting calculation the parcel may be super-saturated.Adjustment is made back to saturation, extra precipitation being added to the rain and snowfalling through the downdraught according to the phase as determined by the saturationtemperature. This amount of precipitation is given by;

The parametrization of evaporation and sublimation is described fully in Gregory (1995). For

(28)∆Pk 1 (q DD DRYk 1 qs(θ

DDsat k 1))

M DDk 1

g

(29)E Lvk 1 A W

v (T evp,p)((108.80(RLρ0.5)0.52) (830.73ρ0.59R 0.67

L ))∆qk 1

∆pk 1

g

rain the rate of evaporation as it falls through layer k-1 is,

while for snow the rate of sublimation in layer k-1 is

where

(30)S LBk 1 A S

v (T evp,p)((1569.52(SLρ0.5)0.55) (32069.02ρ0.63S 0.76

L ))∆qk 1

∆pk 1

g

RL and SL are local rainfall and snowfall rates (defined later) while and (density) is

(31)∆qk 1 (qs(θDD DRY 2k 1 ,pk 1) q DD DRY

k 1 )

estimated using pk-1 and Tevp.

AVW and AV

S are quadratics in temperature which represent the variation of the diffusivity ofwater with temperature and pressure;

36

with p is Pa and T is K.

(32)A W

V (T,p) (2.008x10 9T 2 1.385x10 6T 2.424x10 4) 105

p

A BV (T,p) ( 5.2x10 9T 2 2.5332x10 6T 2.9111x10 4) 105

p

The temperature Tevp used in the evaluation of these quadratics in equations (29) and (30)above is given by,

with θkDD being the potential temperature of the downdraught in layer k after all processes

(33)T evp 0.5(θDDk Πk θDD DRY 2

k 1 Πk 1)

(change of phase, evaporation/sublimation) have been accounted for. Using this temperatureis an attempt to account for the continual variation of temperature due toevaporation/sublimation during descent.

(Note : Version 2A/B of the scheme use an earlier version of the evaporation scheme in

(29a)E Lvk 1 A W

v (T evp,105)(67.08(R Lρ0.5)0.42 541.06ρ0.55R L 0.60)∆qk 1

∆pk 1

g

which the quadratics defined by eqn (32) are not pressure dependent (p being taken to be1000mb=105Pa) and the evaporation rate given by eqn (29) is replaced by;

This form results by assuming a Marshall-Palmer distribution for drop size rather than agamma function of order unity as above.)

It should be noted that eqns (29) and (30) provide local evaporation rates calculated from the

(34)

RL k

Rk

0.5Cconv

SL k

Sk

0.5Cconv

local precipitation rate. However the precipitation calculated in the updraught are grid boxmean rates and so must be divided by the downdraught fractional area to give the local rate.This is poorly defined in the scheme but the assumption is made that the downdraught areais 50% the updraught area. Hence the local rates of precipitation required above are;

37

where Cconv is the convective cloud area calculated in the updraught part of the scheme.

Evaporation and sublimation rates given by eqns (29) and (30) are multiplied by 0.5Cconv toobtain the revised precipitation rates at the lower boundary of layer k-1;

If the rate of evaporation or sublimation exceeds the precipitation rate then EVL and SB

L are

(35)Rk 1 Rk 1 0.5CconvE

Lv

Sk 1 Sk 1 0.5CconvSL

B

set equal to the relevant local precipitation rates and local precipitation rates reduced to zero.

The change in downdraught temperature and mixing ratios due to evaporation and sublimationare,

(36)

∆θDDk 1

g0.5Cconv(LcEL

v (Lc LF)SL

B )

cpΠk 1MDD

k 1

∆q DDk 1

g0.5Cconv(EL

v S LB )

Mk 1

If the evaporation and sublimation produce a supersaturated parcel, the amount of watervapour added to the parcel by these processes is limited by saturation values. Hence if

then

(37)∆θDDk 1≥(θDD

sat k 1 θDD DRY 2k 1 )

and the adjustments to the downdraught mixing ratio and potential temperature increments

(38)

E adj Lv E L

v

(θDDSAT k 1 θDD DRY 2

k 1 )

∆θDDk 1

S adj LB S L

B

(θDDSAT k 1 θDD DRY 2

k 1 )

∆θDDk 1

(eqn (36)) are scaled accordingly;

38

(39)

∆q DD adjk 1 ∆q DD

k 1

(θDDSAT k 1 θDD DRY 2

k 1 )

∆θDDk 1

∆θDD adjk 1 ∆θDD

SAT k 1

(θDDk 1 θDD DRY 2

k 1 )

∆θDDk 1

Precipitation rates, downdraught temperatures and mixing ratios are adjusted accordingly.These represent the final values in layer k-1 after descent of the parcel from layer k.

39

5. Termination of downdraught descent.

The descent of air within the downdraught is driven by the maintenance of negative buoyancyby evaporation/sublimation and water loading. It would therefore seem reasonable that if thedescent becomes positively buoyant at any level after these processes have been accountedfor the downdraught is terminated. However experience with the scheme in single columnmodel tests showed that more continuous descent occurs for deep convective clouds ifdowndraughts are allowed to continue their descent even when they become slightly positivelybuoyant. This may be seen as a crude compensation for the neglect of water loading in theestimation of the buoyancy of the descending parcel

Hence if the buoyancy of the downdraught exceeds 0.5K, i.e.

(40)Bk 1 θDD FINALk 1 (1 Cvirq

DD FINALk 1 ) θE

k 1(1 CvirqE

k 1)>0.5K

then the downdraught is totally detrained into layer k-1 and the descent is terminated.

If this condition is satisfied in the layer below which that from which the downdraughtinitiated then the downdraught calculation is abandoned and the values for the precipitationfalling into layer k-1 adjusted to those which would have been present if no downdraught hadbeen initiated;

where ( )init is the precipitation rate in the downdraught before the calculation of phase

(41)R ENV

k 1 R ENVk 1 (R DD

k 1 )init

S ENVk 1 S ENV

k 1 (S DDk 1 )init

changes and evaporation/sublimation.

If the downdraught penetrates into within 100mb of the surface then terminal detrainment ofthe downdraught occurs according to the formulation described in section 4(a) (eqn (14)).

40

6. Effect of downdraught upon the environment.

This is analogous to the effect of the updraught upon the environment, the downdraughtmodifying the cloud environment through the en/detrainment of heat and moisture and upwardmotion within the clear air compensating the downward motion within the cloud.Consideration of the fluxes of potential temperature and moisture between the cloud and theenvironment in layer k and vertically into and out of layer k (fig 1) shows that the rate ofchange of the θE

k and qEk are given by;

Additional increments are added to the layer in which the downdraught initiates to account

(42)

(∆θE

k

∆t)DD

k

M DDk

∆pk

[(1 εk 1/4∆pk 1/2)(1 ηk∆pk 1/2)(θEk 1 θE

k) ηk∆pk 1/2(θDDk θE

K)]

(∆q E

k

∆t)DD

k

M DDk

∆pk

[(1 εk 1/4∆pk 1/2)(1 ηk∆pk 1/2)(qE

k 1 q Ek ) ηk∆pk 1/2(q

DDk q E

K )]

for the energy required to produce the initial saturated parcel;

In the layer in which the downdraught terminates all the mass is detrained into the

(43)

(∆θE

k

∆t)DD

k (∆θE

k

∆t)DD

k

M DDinit

∆pk

(θDDk θE

K)

(∆q E

k

∆t)DD

k (∆q E

k

∆t)DD

k

M DDinit

∆pk

(q DDk q E

K )

environment;

(44)

(∆θE

k

∆t)DD

k 1

M DDk 1

∆pk 1

(θDDk 1 θE

k 1)

(∆q E

k

∆t)DD

k 1

M DDk 1

∆pk 1

(q DDk 1 q E

k 1)

41

2.7. Treatment of non-downdraught precipitation.

As previously discussed not all the precipitation produced in the updraught falls through thedowndraught. Such precipitation also undergoes changes of phase as it falls and undergoesevaporation and sublimation below cloud base.

a) Change of phase

The state of the non-downdraught precipitation is determined by the temperature of thefreezing level in the cloud environment. Freezing occurs when rain falling from layer k tok-1 crosses the freezing level ( 0oC) or if rain exists in layer k-1 which is below freezing.Melting of snow occurs with the opposite criteria.

The temperature increment averages over the grid box is,

where Pk-1 = RENVk-1 or SENV

k-1 depending upon whether melting or freezing is occurs (as does

(45)(∆θE

k 1

∆t) ±g

LF

cp

Pk 1

Πk∆p DDk 1

the sign).

b) Evaporation/sublimation below cloud base.

Below cloud base it is assumed that non-downdraught precipitation occupies a fractional areaof the grid box equal to the convective cloud area (Cconv) calculated in the updraught part ofthe scheme. The rate of evaporation/sublimination is calculated using equations 29 and 30but with precipitation rates replaced with local rates;

The rate of change of the grid box mean potential and moisture due to

(46)

(R ENVk 1 )L

R ENVk 1

Cconv

(S ENVk 1 )L

S ENVk 1

Cconv

evaporation/sublimation within the area in which precipitation falls are,

42

If evaporation/sublimation is found to cause local supersaturation the rate of

(47)

( ∆θ∆t

)Ek 1 gCconv

(Lc(Ev)L (Lc LF)(SB)

L)

cpΠk 1∆pk 1

( ∆q∆t

)Ek 1 gCconv

((Ev)L (SB)

L)

∆pk 1

evaporation/sublimation are limited so that saturation just occurs by multiplying equations(47) by the factor,

where θESAT k-1 is the saturation temperature of the environment due to evaporation of

(48)Cconv

(θESAT k 1 θE

k 1)

( ∆θ∆t

)Ek 1∆t

sublimination and ∆t is the model timestep.

Rain and snow not evaporated in layer k-1 falls into the next layer or onto the surface.

43

2.8. Energy correction.

As in the original convection scheme (containing an updraught only) a check is made toensure enthalpy is conserved;

where qF is the ice content of the cloud, ∆S is the increment due to changes of phase only,

(49)cp⌡⌠pT

p( ∆T

∆t)dp Lc⌡

⌠pT

p( ∆q

∆t)dp LF⌡

⌠pT

p(

∆sqF

∆t)dp

and pT is the pressure at the top of the model.

The second term on the rhs was previously determined according to the temperature of thelowest model layer as this determined the state of the precipitation at the surface. Howeverthe inclusion of the downdraught allows snow to exist at the surface when the lowest modellayer temperature is greater than 0oC. Hence if,

or

if Ssurf > 0 (50a)LF⌡⌠

p

pT∆sqF

∆tdp gLFSsurf

Hence,

if Ssurf = 0 (50b)LF⌡⌠

p

pT∆sqF

∆tdp 0

Note that ∆T/∆t and ∆q/∆t here refer to the total effect of the updraught and downdraught.

(51)cp⌡⌠pT

p( ∆T

∆t)dp Lc⌡

⌠pT

p( ∆q

∆t)dp gLFmax(0,SSURF)

Due to errors in the discretisation of cloud model this balance is not exact and,

(52)cp⌡⌠pT

p( ∆T

∆t)dp Lc⌡

⌠pT

p( ∆q

∆t)dp gLFmax(0,SSURF) RES

The residual on the rhs is reduced to zero as in the original version of the scheme (1A) bythe adjustment of those temperature increments which are the same sign as the residual.

Once the increments have been adjusted to conserve enthalpy the model potential temperatureand mixing ratio are updated to account for the effects of both updraught and downdraught.

44

References for part 2

Gregory, D., 1995 : Convection scheme, part 1 of UM Documentation Paper 27 (usuallyattached and preceding the current part), 27 available from the UM Librarian, NWP-Div.,Meteorological Office, London Road, Bracknell, Berks. RG12 2SZ

Gregory, D. and Allen, S., 1991 : The effect of convective scale downdraughts upon NWPand climate simulations., Preprints of the 9th conference on Numerical Weather Prediction,Oct 14-18, 1991, Denver, Co., USA.

Gregory, D., 1995 : A consistent treatment of the evaporation of rain and snow for use inlarge-scale models., Mon. Wea. Rev., 123, No. 9, pp 2716-2732

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Part 3 - Diagnosis of Deep and Shallow ConvectionTracer TransportsMomentum TransportsCAPE closureInitialisation of Convection by Surface Fluxes

CONTENTS

3.1. Introduction

3.2. Diagnosis of deep and shallow convection

3.3. Tracer transports within convection

3.4. Momentum transports within convection - TEST VERSION

3.5. Convective Available Potential energy closure- TEST VERSION

3.6. Initialisation of convection by surface fluxes

REFERENCES for part 3

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3.1. Introduction

Since a convective downdraught was introduced into the UM convection scheme as version2A (described in documentation paper 27-2), there have been several other modifications tothe convection scheme. These changes will be described in this paper.

The current scheme does not explicitly differentiate between deep and shallow convection,but instead uses a bulk cloud model which attempts to parameterize the effects of anensemble of different types of convective clouds (see documentation paper 27-1). The firstmodification, which is described in section 3.2 of this paper, introduces a scheme whichdetermines whether convection within a grid box will be deep or shallow, and allows differentconvective parameters to be assigned for the convective calculations depending on the resultsof this diagnosis. This diagnosis is only carried out in the 3A version of the convectionscheme, introduced at version 4.0 of the Unified Model as a test version. Currently, deep,shallow and mid-level convection are treated in exactly the same way and the tests are simplydiagnostic.

The second modification to the convection scheme described in this paper allows atmospherictracers such as anthropogenic gaseous emissions, volcanic emissions etc. to be transportedvertically within convective clouds, both in the updraught and the downdraught. There iscurrently no treatment of tracers being washed out by precipitation. This too is only includedin version 3A of the convection scheme.

A parametrization of the effects of vertical transports of momentum has been included in theconvection scheme. The treatment has been validated against results derived from cloudresolving model simulations. Such transports play an important role in the distribution ofmomentum in the vertical within convectively active regions, particularly within the tropics.Redistributing momentum within the tropics will also affect the transports of momentum intothe sub-tropics and mid-latitudes by convectively driven circulations such as the Hadley Celland the Indian monsoon, and so will affect the entire global circulation pattern of the model.This scheme is currently included as a test version only, controlled by a switch in the 3Aversion of the convection scheme.

The closure of the convection scheme described in documentation paper 27-1 is based on thelocal buoyancy of a parcel lifted over a vertical interval. Section 3.5 of this paper willdescribe an alternative closure based on the convective available potential energy (CAPE) ina vertical column of the atmosphere. In this scheme, the closure is determined by calculatingthe initial convective mass flux necessary to reduce the CAPE to zero over a given timescale.Once again, this scheme is included as a test version only, controlled by a switch in the 3Aversion of the scheme.

In versions 1A and 2A of the convection scheme, the initial increments to potential

47

temperature and humidity mixing ratio of the convective parcel are prescribed as being 0.2Kand 0.0kg/kg respectively. In the 2B version of the scheme, for convective parcels initiatingin the lowest model layer, the initial increment is diagnosed from the turbulent fluxes oftemperature and moisture calculated by the boundary layer scheme. This has a potentiallylarge effect on the initial buoyancy of the parcel in areas where the surface fluxes of heatand/or moisture are large, such as over desert areas or the trade wind regions of the tropicaloceans.

3.2. Diagnosis of deep and shallow convection

The Unified Model uses a convective parametrization which attempts to represent the effectsof an ensemble of different types of convection within each grid box. However, manyconvective regions are dominated by either shallow convection which initiates and terminateswithin the planetary boundary layer, or deep convection which penetrates into the freeatmosphere, often extending through the depth of the troposphere. These two types ofconvection interact differently with the environment through entrainment and detrainment.Deep convective plumes tend to be fairly undilute, whereas shallow convection tends to bemore turbulent, with greater mixing of environmental air into the cloud, which limits thevertical extent of the convection by reducing the buoyancy of the convective parcels. Theoriginal Unified model convection scheme attempts to parameterize these differingcharacteristics by assigning an entrainment rate which varies linearly with pressure. This isdescribed fully in Documentation Paper 27, section 1.2(c). The result is that at low levelswhere deep and shallow convection can exist together, the entrainment rates will be high,whereas higher in the atmosphere where only deep convection exists, entrainment rates arelower so that there is less mixing of air between the convective plumes and the environment.

The modified scheme performs a preliminary test ascent in order to determine whether deepor shallow convection will develop within the grid box. It is then possible for the scheme toassign different entrainment rate coefficients and mass flux parameters based on thisdiagnostic ascent before performing the full convective calculations. For grid points on modellevels within the boundary layer, convection is initiated as in the 2B version of the scheme(see section 3.6) and then if the parcel is buoyant in layer k, it is lifted to layer k+1 withentrainment processes taken into account as normal. The parcel potential temperature andmixing ratio in layer k+1 are given by first undergoing a dry ascent.

48

Moist processes are then taken into account and if the parcel is saturated in layer k+1 thenthe parcel values of θ and q are recalculated.

Where L = latent heat of condensation (or condensation + fusion if the water is frozen)Πk+1 = (pk+1/100000) and = R/cp

Details of the calculation of the condensation term CONDk+1 are given in documentation paper27-1.

The buoyancy of the parcel in layer k+1 is calculated and the ascent continues until the parcelbecomes negatively buoyant or reaches the top of the boundary layer. Parcel buoyancy isgiven by,

where the v subscript denotes virtual potential temperature and the E superscript denotesenvironment values.

If the buoyancy becomes less than or equal to zero within the boundary layer then shallow

49

convection is said to be occurring and a logical switch for shallow convection is set to true.If the parcel is still buoyant above the boundary layer then deep convection will occur andthis logical switch remains set to false. A third possible case is that convection initiates abovethe boundary layer in which case mid-level convection is said to be occurring, and a logicalswitch for mid-level convection is set to true.

Documentation paper 27 describes how, if a parcel becomes negatively buoyant in layer k+1,forced detrainment can bring the parcel back to positive buoyancy in layer k+1 thus allowingthe ascent to continue. The forced detrainment calculations are not performed in this testascent as they are computationally expensive. However, as stated in 27 section 1.6, the upperlimit for ascent of a parcel is the point at which an undilute parcel from the start level ofconvection would become negatively buoyant. The virtual potential temperature of such anundilute parcel is calculated using equation 32 from Paper 27, and if

(where XSBMIN = minimum parcel excess buoyancy to continue ascent)in the top 2 levels of the boundary layer then the parcel is assumed to be still buoyant andthe switch for shallow convection is reset to false.

The state of the logical switches then determines the choice of mass flux parameter, initialbuoyancy and entrainment rates for the full dilute ascent which is subsequently carried outas described in paper 27-1. The scheme begins by assigning deep convective values of massflux parameters, parcel initial θ and q increments and entrainment rate coefficients, or mid-level values if the convection initiates above the boundary layer. These are then reset toshallow convective values if the test ascent terminates within the boundary layer. Currently,the entrainment rate coefficients for deep convection are set using the same formulation asin the original scheme;

from equations 14 in paper 27-1.

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The code allows shallow convective entrainment and detrainment to be multiplied by a factor

Currently FAC is set to 1 retaining the original entrainment formulation. At present, valuesof entrainment and detrainment for mid-level convection are identical to those for deepconvection.

3.3. Atmospheric tracer transports in convection

This section describes the vertical movement of atmospheric tracers within convectivesystems. Tracers are passive variables in that they are moved up and down within convectiveupdraughts and downdraughts without affecting the evolution of the model convection in anyway. There is also currently no treatment of the removal of tracer from the atmosphere byconvective precipitation.

(a) Tracer in the updraught

When convection within a grid box is initiated, the initial value of tracer in the ascendingparcel is set to the grid box environment tracer mixing ratio at the level at which convectioninitiates.

The parcel then undergoes convective ascent from layer k to k+1 as described in 27-1 andthe parcel tracer mixing ratio in layer k+1 is given by,

51

where

The values of entrainment coefficients ε are the same as those used for mixing of θ and q.The ascent continues until the conditions for termination of convection described in 27-1 aremet.

The tracer mixing ratio of the cloud environment is modified by convection throughcompensating subsidence, forced and mixing detrainment. The rate of change of environmenttracer mixing ratio is given by,

where Mk = massflux in layer k∆pk = layer thicknessδk = parcel forced detrainment rate

52

µk = parcel mixing detrainment rate

The three terms in square brackets represent compensating subsidence, forced detrainment andmixing detrainment. This is analogous to equations 38a and 38b in 27-1 for environment θand q, but there are two differences. Firstly, forced detrainment of tracer occurs at the parcelvalue of tracer mixing ratio rather than an adjusted parcel value (represented by θR and qR

in 27-1). Secondly, there is no term to include the effect of the detrainment of cloudcondensate as this does not affect tracer amounts, although a full treatment of the interactionof tracers with cloud water and precipitation might do.

Terminal detrainment of tracer at the top of the convective plume occurs in the same way asterminal detrainment of θ and q as described in section 1.10 of 27 i.e. a proportion of thetracer in the convective parcel is also detrained into the model layer above that at whichconvection actually terminates. So on the termination of convection,

(b) Tracer in the downdraught

When a convective downdraught is initiated as described in Documentation Paper 27 part 2,the initial value of tracer mixing ratio in the downdraught is assumed to be an equal mixtureof the cloudy tracer mixing ratio and the tracer mixing ratio of the surrounding clear air.Thus,

The parcel then undergoes descent from layer k to k-1 as defined in 27 section 1.4, with

53

mixing entrainment and detrainment occurring so that the tracer mixing ratio in thedowndraught in layer k-1 is given by,

Values of ε are the same as those used for entrainment of θ and q as described in 27-2,equation 11 and 12. Downdraught descent is terminated as described in section 2.5 of Paper27.

The tracer mixing ratio of the cloud environment is modified through the detrainment oftracer from the downdraught and the upward motion of clear air to compensate for thedownward motion in the cloud. The rate of change of tracer mixing ratio in layer k due to theconvective downdraught is given by,

where ηk = downdraught mixing detrainment rate. (see Paper 27, section 2.4a, equations 13and 14.)

The first term on the RHS represents upward motion of cloud-free air to compensate for the

54

downdraught, and the second term represents mixing detrainment. In the model layer at whichthe downdraught terminates, all the tracer mass is detrained into the environment.

An additional adjustment is made to the tracer content of the layer in which the downdraughtinitiates to take into account the tracer used to form the initial downdraught parcel.

3.4. Momentum transports within convection

The current treatment of convective momentum transports is based on the above descriptionof tracer transports, i.e. momentum is treated as a passive tracer which does not directlyimpact upon the development of convection within a grid box at any one timestep. However,there will clearly be an indirect feedback as momentum transport will modify the horizontalflow pattern which will then change the patterns of temperature and moisture advection. Thusthe stability of the atmosphere will be modified and the pattern of convection will thenchange.

(a) Momentum in the updraught

Since momentum values are stored in the model on the UV grid, whereas convectioncalculations are carried out on the P grid, the winds must be interpolated onto the P gridbefore momentum values can be assigned to convective parcels. This is done in the controlstructure of the model. When convection within a grid box is initiated, the parcel is assignedvalues of momentum equal to the values of zonal and meridional windspeeds in the grid boxat the level at which the convection starts.

The parcel undergoes convective ascent from layer k to k+1 as described in 27-1 and the

55

parcel values of u and v are modified by entrainment of air with the environmental values ofmomentum. The parcel value of u in layer k+1 is given by,

and similarly for v.

Once again, the values of the entrainment coefficients used in this calculation are the sameas those used for the entrainment of θ and q. The parcel ascends until the buoyancy criterionfor the termination of convection is met.

For deep convection, evidence from detailed cloud modelling studies suggests that across-cloud pressure gradients greatly modify in-cloud velocities, bringing them closer toenvironmental values than would be the case by entrainment alone (Gregory and Miller,1989). Cloud modelling studies by Gregory et al (1995, in preparation) suggest that these maybe approximated by,

where C is a constant.

This formulation is purely empirical. The constant is chosen by comparison with the termdiagnosed from detailed cloud models and is currently set to 0.7 (Gregory et al. 1995). Thisterm is applied at level k+1/2.

Hence u in layer k+1 is finally given by

56

where [upk+1]ent is given by eqn(17)

Note that this pressure gradient correction is only applied in cases of deep convection asdiagnosed by the method described in section 3.2 above.

A different but equivalent formulation is used to calculate the impact of convection upon thelarge-scale momentum field. This is given by,

and similarly for v.

It is possible to write this in terms of subsidence and detrainment plus a cloud pressuregradient term, as for heat and moisture, but this is inconvenient given the structure of thecode. Hence (20) is discretised directly using an upstream difference approach;

where

57

and

whereMk = updraught massfluxδk = parcel forced detrainment rateµk = parcel mixing detrainment rate

At the starting level of convection, Fk-1/2 = 0, while in the final detraining layer, Fk+1/2 = 0.

A similar approach is applied to v.

It should be noted that in the actual code, Fk-1/2 is not calculated as above, but Fk+1/2 is storedand passed through the loop over model levels, becoming Fk-1/2 as k increases by 1.

(b) Momentum in the downdraught

The initiation of a convective downdraught remains exactly as described in documentationpaper 27-2. The initial values of zonal and meridional momentum in the downdraught parcelare assumed to be an equal mixture of the momentum in the terminating updraught and thatin the cloud environment. So,

with a similar equation for the initial value of v.

The parcel then undergoes descent from layer k to k-1 as defined by Paper 27, section 2.4.The parcel values of u and v are modified by entrainment and mixing detrainment, so themomentum of the parcel in layer k-1 is given by

58

The values of entrainment coefficients ε are the same as those used for entrainment of θ andq.The value of u in the parcel is then further modified by the in-cloud pressure gradient termdescribed above.

With a similar equation for v. C is set to 0.7 as for the updraught.

The zonal and meridional winds in the cloud environment are modified using the same typeof eddy flux formulation described above for the updraught.

where

andMk = downdraught massfluxηk = downdraught mixing detrainment rate

Fk+1/2 is not calculated directly, but Fk-1/2 is stored and passed through the loop over modellevels, becoming Fk+1/2 as k decreases by one count. In the layer at which the downdraught

59

initiates, Fk+1/2 = 0, and in the layer at which the downdraught terminates, Fk-1/2 = 0.

An additional adjustment is made to the momentum in the layer in which the downdraughtinitiates to take account of the momentum used to form the downdraught parcel

with an equivalent equation for v in the initial layer.

3.5. CAPE Closure

In convection schemes which use an adjustment type closure, the initial cloud base mass fluxis calculated from relaxing the atmosphere back to an equilibrium state. In the CAPE closurescheme described here, the cloud base mass flux is calculated based on the reduction to zeroof convectively available potential energy (CAPE) over a given timescale. This closure isbased on that of Fritsch and Chappell (1980).

CAPE is defined as

Assuming steady state clouds, the rate of change of CAPE with time due to convectiveactivity is given by

60

Discretizing this equation, converting to potential temperature rather than virtual potentialtemperature and using a reduction of CAPE to zero over a timescale τ gives,

The rates of change of θE and qE with time are calculated by the model, but requireadjustment to give the correct rate of dissipation of CAPE. Writing rates of change of θ andq in terms of the cloud base mass flux,

(where Q1’ and Q2’ indicate rates of change of θ and q calculated by the model, scaled bythe original cloud base mass flux calculated using the original closure, see Paper 27, section1.5.)

Substituting into equation 31 allows the adjusted cloud base mass flux to be calculated.

The rates of change of θ and q calculated by the model are multiplied by MBnew/MB to give

values which dissipate CAPE at the correct rate.

3.6. Initialisation of convection by surface fluxes

In the original version of the convection scheme (1A and 2A), the initial increments to parceltemperature and moisture are prescribed (0.2K and 0.0g/kg respectively). In versions 2B and3A of the scheme, for convection initiating from the lowest model layer, these increments are

61

initiated from surface fluxes of heat and moisture, and are based upon the standard deviationof T and q fluctuations associated with boundary layer turbulence.

The standard deviations of the turbulent fluxes of temperature and moisture at the lowestmodel level (T1SD and q1SD) are passed into the convection scheme and the initial parcelvalues are given by

Where 0.2 and 0.0 are the initial convective increments to θ and q respectively, prescribedby the original convection scheme.

REFERENCES for part 3

Fritsch, J.M. and Chappell, C.F., 1980 : Numerical prediction of convectively drivenmesoscale pressure systems. Part I : Convective parameterization., J. Atmos. Sci., 37. 1722-1733.

Gregory, D. and Miller, M.J., 1989 : A numerical study of the parameterization of deeptropical convection., Quart. J. Roy. Meteor. Soc., 115. 1209-1242.

Gregory, D., Kershaw, R. and Inness, P. M., 1995 : Parameterization of momentumtransport by convection. II : Tests in single column and general circulation models. Inpreparation.

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Part 4 - anvil scheme

4.1. Introduction

The Anvil scheme was introduced at version 4.4 of the UM and is intended to represent theradiative effects of convective anvils. It does this by specifying a cloud amount that variesin the vertical, rather than being constant. It is an option to versions 3A and above ofconvection, and must be used with the Edwards-Slingo radiation scheme.

4.2. Calculation of the new cloud fraction.

The convective cloud amount (as calculated in equation 37) is expanded into the vertical andmodified under certain conditions to produce a cloud amount that varies with height. If thecloud base is in the boundary layer, the cloud top is above the freezing level and the cloudis diagnosed as ’deep’ then an anvil is applied. A ’deep’ cloud is one that is more than 500mbbetween top and base. However this depth criterion may be switched off for modelintegrations so there is no depth constraint on the clouds.

The anvil base is defined as the freezing level. In cases where the entire cloud is above thefreezing level (eg. as might occur in mid-latitude winter) the anvil base is said to coincidewith the cloud base. i.e.

The cloud is divided into two portions: the anvil and the tower. The cloud amount in thetower is given by:

while the cloud fraction in the anvil is:

where CCA = convective cloud amount varying with model levelCCA2D= convective cloud amount on a single level (eqn 37)

TF = tower factorAF = anvil factork = model level. Subscripts cct, ccb, freeze and anv refer to cloud top, cloud base,

freezing level and anvil base respectively.

63

If the cloud is not diagnosed as deep, then the old cloud fraction, CCA2D, is kept, but isexpanded onto model levels.

64