The sensitivity of simulated high clouds to ice ... - Wiley Online Library

Quarterly Journal of the Royal Meteorological Society

Q. J. R. Meteorol. Soc. 141: 1546–1559, July 2015 A DOI:10.1002/qj.2457

The sensitivity of simulated high clouds to ice crystal fall speed, shape and size distribution K. Furtado,* P. R. Field, R. Cotton and A. J. Baran Met Office, Exeter, UK *Correspondence to: K. Furtado, Met Office, FitzRoy Road, Exeter, EX1 3PB, UK. E-mail: [email protected] This article is published with the permission of the Controller of HMSO and the Queen’s Printer for Scotland.

The sensitivity of an operational numerical weather prediction model to the parametrized microphysical properties of ice hydrometeors is examined. The effects of varying ice-particle size distribution, fall speed, mass and depositional capacitance are considered in kilometrescale simulations of midlatitude cloud systems and 20 year global climate integrations. It is shown that the observed sensitivity can be obtained from steady-state arguments, whereby the vertical moisture flux is balanced by the hydrometeor sedimentation flux and supersaturation production is in equilibrium with depositional growth. The high-resolution simulations are compared with in situ measurements from the Constrain field campaign (Prestwick, UK). Key Words: Unified Model; cirrus; ice microphysics Received 8 April 2014; Revised 28 July 2014; Accepted 1 September 2014; Published online in Wiley Online Library 19 November 2014

1. Introduction Motivated by the role of high clouds in weather and climate, several previous studies have addressed aspects of the sensitivity of simulated clouds to ice microphysics (e.g. Wu et al., 1999; Grabowki, 2000, 2003; Wu, 2002; Iga et al., 2007; Satoh and Matsuda, 2009; Molthan and Colle, 2012; Satoh et al., 2012; Th´eriault et al., 2012; Han et al., 2013; Weverberg et al., 2013). Molthan and Colle (2012) and Han et al. (2013) examined the effect of the choice of cloud microphysics scheme on Weather Research and Forecasting Model (WRF) simulations of midlatitude cyclones and found significant ice cloud differences. Weverberg et al. (2013) simulated tropical mesoscale storms, again with WRF, and looked at the sensitivity to a subset of the schemes considered by Molthan and Colle (2012). The simulated tropospheric humidity structure varied between the schemes and this was found to be related to differences in the cloud condensed water contents and mean ice sedimentation velocities. In simulations with a small-planet general circulation model (GCM), Satoh and Matsuda (2009) changed ice particle fall speeds and size distributions and found that decreasing the fall speed increased high cloud cover. The studies of Iga et al. (2007, 2011) and Satoh et al. (2012) found the same cloud response to similar sensitivity tests, performed with a GCM (as did the earlier cloud-resolving studies by Wu et al. (1999) and Wu (2002)). Grabowki (2000, 2003) varied the size distributions in cloud-resolving, radiative–convective equilibrium simulations and found that higher concentrations of small particles led to a warmer troposphere that was closer to water saturation. Similarly, Iga et al. (2011) and Satoh et al. (2012) found that increased GCM cloudiness was associated with increased relative humidity in the subsaturated mid-troposphere.

We confine ourselves to a single scheme, that of Wilson and Ballard (1999), and make a sequence of changes to parametrizations within that scheme. The influence of various combinations of changes can then be isolated. For example, effects due to the assumed particle size distribution (PSD) can be separated from those arising from particle fall speed. We consider how cirrus clouds are affected when the size distributions, fall speeds, masses and shapes of ice hydrometeors are changed. The model sensitivities are performed with the Met Office Unified Model but to interpret them we will develop a framework which should be generally applicable to other models. We begin with an analysis of three high-resolution case studies of midlatitude cirrus and then consider the impact of microphysics changes on 20 year integrations of the global climate. The highresolution simulations are evaluated against in situ measurements made by the Facility for Airborne Atmospheric Measurements (FAAM) BAe146 research aircraft. Prior to the advent of complex microphysics for numerical weather forecasting, Passarelli (1978) used steady-state models to study the vertical structure of snowstorms. In steady-state conditions, the upward flux of total water is balanced by the downward flux of ice crystals and the snowfall rate is a function of the vertical velocity profile (Passarelli, 1978; Mitchell, 1988). Steady-state solutions are useful because they exhibit explicitly the dependence of the vertical cloud structure on microphysical parameters. Hence it is easy to understand how steady states respond to changes in microphysics. Moreover, it is tempting to invoke implicit steady-state assumptions when reasoning heuristically about the microphysical sensitivity of clouds. For example, consider the ice supersaturated region, close to cloud top, in a midlatitude storm system: it seems intuitive that increasing the ice particle fall speed will result in less ice and

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High Cloud Sensitivity to Ice Size and Fall Speed higher humidity aloft. However, in a dynamical situation, a feedback results whereby the increased supersaturation leads to enhanced ice growth. One may then question whether the resultant rate of increase of ice water content is sufficient to counterbalance the sedimentation rate. This kind of circular reasoning can be avoided by looking only at time-independent states in which sedimentation, depositional growth and moisture transport are in equilibrium. However, it is not obvious that steady-state models are applicable to real clouds, or to dynamic simulations of real clouds. We will examine the extent to which average simulated cloud structures are steady. We will then show that the effects of microphysics changes can be understood in terms of simple steady solutions to the model equations. The main motivation for studying the effects of microphysics parametrizations is to determine optimal configurations for weather forecasting and climate modelling. We therefore evaluate the results of sensitivity tests against ice water content (IWC) and PSD measurements made by the FAAM BAe146 during the Constrain field campaign (Cotton et al., 2012), which allows us to draw some conclusions about the performance of the parametrizations tested. The seamless approach to weather and climate modelling is predicated on using the same model physics across scales. It is therefore useful to understand how parametrizations that have been tested against high-resolution observations effect longterm climate statistics. Moreover, the radiative impact of cirrus depends strongly on microphysical modelling assumptions. For example, changes that reduce the amount of high cloud increase the transmission of terrestrial radiation (Wu et al., 1999; Wu, 2002; Iga et al., 2007, 2011) and reduce the reflected solar flux (Wu et al., 1999). Such differences can significantly affect tropospheric temperatures (e.g. Wu et al., 1999; Grabowki, 2000, 2003; Wu, 2002; Edwards et al., 2007). We compare one of the microphysics configurations from the high-resolution tests with a control model that has different PSD, mass–diameter and fallspeed parametrizations. We show that the climate differences can be understood in terms of the high-resolution sensitivities. 2. Cases We consider three high-resolution modelling case studies. The simulations correspond to three separate occasions in January 2010 when the FAAM BAe146 aircraft flew sorties out of Prestwick in western Scotland. Two of the sorties, referred to as flights b500 and b503, took place in frontal cirrus over the North Sea. A third sortie, flight b497, took place in orographically forced ice cloud occurring over the Scottish Highlands. Figure 1 shows simulated ice water path (IWP) fields for the three cases and shows the geographical locations of the cloud systems relative to the United Kingdom. The solid white curves show the aircraft flight paths for each sortie. The sorties were so-called ‘Lagrangian descents’, where the aircraft drifted with the mean horizontal wind whilst descending at a constant rate chosen to be comparable to the estimated mean fall speed of the ice particles (approximately 0.5 m s−1 ). 3. Models The high-resolution domains were nested directly inside an N512 global model (25 km midlatitude grid spacing). The highresolution domains had an approximate grid spacing of 1.5 km (zonal and meridional angular spacing of 0.0135◦ ). The horizontal extent of the domains was 1200 km in the north–south and east–west directions. The models had 118 levels in the vertical, with spacing varying from 5 m near to the surface to 3 km at 78 km (with 0.3 km representative of 10 km). The time step was 10 s and the local area lateral boundary conditions were updated hourly. The forecast lengths were 18 h, starting at 0000 GMT on the day of each case. The vertical level set and time-stepping differ

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from those used operationally, as do the specific settings in some of the subgrid physics schemes. The model is non-hydrostatic and uses a semi-Lagrangian advection scheme. No convection parametrization is used. The boundary-layer parametrization is the non-local scheme described by Lock et al. (2000) and operates in the lowest 30 model levels (below 3 km). The band-averaged cirrus bulk optical properties are obtained from the ice-crystal ensemble model described in Baran et al. (2014). The cloud microphysics scheme is a single-moment scheme with prognostic fields for cloud droplets, rain, graupel and snow. The snow field includes all frozen hydrometeor mass, except graupel. The following aspects are relevant for ice cloud simulation. Ice can sediment, grow by deposition and accrete mass from other hydrometeor species. Ice is nucleated by homogeneous freezing below −40◦ C and by heterogeneous nucleation below −10◦ C. Ice-cloud fractions and subgrid moisture variability are diagnosed using the cloud scheme documented by Wilson et al. (2008). The critical relative humidity parameter, RHc , is set to 0.99. This implies that the ice-cloud fraction, φ , is approximately either one or zero in each grid box. 3.1.

Ice microphysics parametrizations

We will assess the impact of changing the parametrizations of ice fall speed, mass, PSD and depositional capacitance. In this section we describe the different parametrizations used. In section 3.2, we state the ice microphysical configurations of the sensitivity tests. 3.1.1.

Mass–diameter parametrizations

Microphysics schemes need mass–diameter (m–D) relations to obtain closed expressions for the assumed particle size distributions and hence for microphysical process rates. The ice-crystal m–D relations used in this study are power laws: m = aDb ,

(1)

where the ‘diameter’ D is some measure of the linear size of the ice crystal. Many different relations have been proposed and these are often dependent on ice crystal habit and have different ranges of validity. The m–D relations used for the model sensitivities are specified in Table 1. These are taken from literature sources and have been chosen because they have been used in various Met Office operational forecast models. Two of the m–D relations are taken from Mitchell (1996, his table 1). These were used to represent the masses of ice crystals and snow aggregates in the operational Met Office UKV until January 2012. We also test the relation from Brown and Francis (1995) that is currently used in the Met Office Global operational forecast model. Another relation used is from Cotton et al. (2012). This relation was derived from the Constrain in situ data and is therefore the relation that most accurately describes the cirrus clouds in the cases studied in this article. Of these three m–D relations, only the relation from Cotton et al. (2012) was derived from data obtained using technology designed to reduce the shattering of ice crystals on the aircraft instrumentation. 3.1.2.

PSD parametrizations

The particle size distribution, f (D), gives the number density of particles per unit size. The moments of the PSD, Mn = Dn f dD, n ≥ 0, determine microphysical process rates used to predict cloud evolution. Parametrizations can treat the PSD as composed of one or more submodes, each of which contains part of the total IWC. For each mode, the PSD and IWC are related by the mode m–D relation. Two PSD parametrizations are considered in this study: a parametrization based on Houze et al. (1979) (see also Cox, 1998) that splits the PSD into modes and a more


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K. Furtado et al.

(a)

(b)

(c)

recent approach due to Field et al. (2007) that uses a single PSD to describe all particle sizes. We will refer to the modal PSD as the diagnostic-split PSD parametrization because the total IWC is diagnostically divided between two PSD modes, referred to as ‘snow’ and ‘ice’. The snow mode has lower number concentration and represents the distribution of larger, more aggregated particles. The ice mode has higher number concentration and represents smaller, more pristine crystals. The IWC fraction in the snow mode is an empirical function of the temperature difference, T, from the local cloud top and is described in detail by Cotton et al. (2012). The two modes have exponential PSDs of the form N0 exp(−λD), where λ is determined from the mode IWC. The m–D relations for the two modes are typically chosen so that, for a given IWC and T, the ice-mode is narrower than the snow-mode. The intercept functions, N0 , are exponential functions of the temperature, T, and are different for ice and snow. The intercept parameter for snow, N0a , is based on the study by Cox (1998), which used the Houze et al. (1979) data set. The intercept parameter for ice, N0i , is arbitrarily set to 20N0a (Wilkinson, 2013). The underlying dataset from Houze et al. (1979) contains a small number of measurements (of the order of 40), confined to temperatures warmer than −45◦ C. Moreover, the data pre-date attempts to understand and account for the effects of ice-crystal shattering. The second PSD parametrization tested is a so-called moment estimation parametrization, due to Field et al. (2007). A single PSD is used which contains all the ice crystals and has an underlying ‘universal’ form when non-dimensionalized. The PSD moments were parametrized empirically based on a large dataset and are related to the second moment, M2 , by equations of the h form Mn = gn M2n , where gn , hn are functions of temperature. The moment relations were derived using only particle sizes D > 100 μm and therefore do not include the uncertain small particle contribution. Moreover, the PSD was derived from data that had been corrected, as best as possible, for the effects of shattering. 3.1.3.

Fall speed parametrizations

Ice crystal fall speed, Vt , and size are related by power laws: Vt = cDd .

(2)

Fall speeds are used to calculate sedimentation rates and other processes that depend on particle velocity, e.g. riming and

Vt (m s–1)

Figure 1. Examples of model IWP fields for each of the simulated cases: (a) b500 and (b) b503, the two cases of frontal cirrus approaching the United Kingdom (coastline shown in white) from the west, and (c) b497, the case of orographic cirrus over the Scottish Highlands. The large grey boxes show the averaging areas used for a spatially averaged budget analysis of the cloud systems. Note that for the orographic case the ‘large’ area is actually quite small and centred on the Scottish Highlands to avoid including the cumulus clouds present over the ocean. The smaller grey boxes show the averaging areas used when the model fields are compared with in situ aircraft observations. The flight paths followed by the FAAM BAe-146 aircraft on each day are shown by the coil-shaped white curves.

diameter (µm) Figure 2. Fall-speed–diameter relations. For the diagnostic-split PSD experiment (X), the black–white dashed line shows the snow fall speeds and the thick black line shows the ice fall speeds. The black–white dashed line is also the relation for experiment Z and the snow fall-speed relation for the dual fall-speed experiment, W. The solid white line shows the ice fall speeds for experiment W. The dashed white line is the relation for experiment Y. The thin grey lines show the data from Mitchell (1996).

depositional growth. The Vt –D relations used in this study are plotted in Figure 2, together with measurements from Mitchell (1996). We have chosen relations that sample a realistic variability range and have been used in operational Met Office models. The diagnostic-split PSD can have different values of c and d for the ice and snow modes. Sedimentation fluxes are calculated for each mode and added together to obtain the total mass flux. The typical situation is that the ice-crystal fall-speed relation gives slower Vt for small values of D than the snow relation. At some size, the two relations cross over and ice particles become the faster falling species. Examples of this behaviour can be seen in Figure 2. The freedom to specify different Vt –D relations for the snow and ice helps represent the variability of the measurements. For example, as can be seen from Figure 2, power laws representative of large aggregates tend to overestimate the fall speeds of small crystals. Consequently, it is useful to be able to use a combination of power laws when calculating sedimentation fluxes, even when the PSD is not decomposed into separate ice and snow modes. The moment-estimation PSD is usually applied with a single Vt –D relation, but this is not an intrinsic limitation. For one of the experiments we implement the following method, which effectively allows small and large particles to sediment


Q. J. R. Meteorol. Soc. 141: 1546–1559 (2015)

High Cloud Sensitivity to Ice Size and Fall Speed with different Vt –D relations. First, two fall-speed relations are specified; for consistency with the diagnostic-split PSD nomenclature, we will call these the relations for ice and snow. These relations are chosen so that ice is the slower falling species at small sizes (e.g. for D 600 μm) and the faster falling species for large sizes. Next, given the two Vt –D relations, two candidate massweighted mean fall speeds are calculated in each model grid box using the moment-estimation PSD. The set of fall-speed parameters that gives the least mean fall speed is then selected and used to calculate the sedimentation rate and other fall-speeddependent processes. For example, if the mean fall speeds for ice and snow in a particular grid box are Vm,i and Vm,a , then the mean fall speed used will be Vm = min{Vm,i , Vm,a }. This method therefore blends between the two fall-speed relations as the PSD evolves. Although using a minimum condition on the mean fall speeds implies that there is no explicit ‘size threshold’ below which the blending uses the ice fall-speed parameters, the transition will occur approximately where the mean particle size reaches the size at which the ice and snow fall-speed curves cross each other (i.e. around 600 μm for the examples in Figure 2). In grid boxes where the PSD is narrow and the mean particle size relatively small, the ice Vt –D relation will determine the sedimentation flux, because the mean fall speed is weighted towards small sizes, where the snow terminal velocity exceeds that of ice. On the other hand, when the PSD is broad the mean fall speed is weighted towards large sizes and the snow relation will determine the sedimentation flux. Selecting the minimum of the two candidate mean fall speeds ensures that the sedimentation flux is a continuous function of IWC. The terminal velocity parameters, however, are likely to change discontinuously between regions where the sedimentation uses the ice and snow parameters. As such, the method represents a somewhat pragmatic approach to allowing the observed fall speed variability with a single ice category. We will call this approach the dual fall-speed parametrization. 3.1.4.

Depositional growth parametrization

We use the standard parametrization for crystal growth, whereby the rate of change of mass of an individual crystal is m ˙ = Cd B0 DFv S,

(3)

where B0 is an effective water-vapour diffusivity, Fv is a factor accounting for ventilation effects and the supersaturation, S, is defined by S = ρv /ρs − 1, where ρv is mass density of water vapour and ρs is its ice-saturated value (see Appendix A for complete definitions). The constant Cd is the depositional capacitance, which attempts to incorporate the effects of ice crystal shape into the growth-rate parametrization. An idealized spherical ice crystal would have Cd = 1, whilst more plate-like crystals have capacitance Cd ∼ 1/2. The bulk growth rate of the particle population is found by ˙ over the PSD. The presence of ice particles causes integrating m the ice supersaturation to relax to S = 0. The characteristic timescale for this process is the phase-relaxation time-scale (Korolov and Mazin, 2003):

τp =

1 , Cd β B0 Mv1

(4)

where β is a function of pressure and temperature and Mv1 = DFv f dD is the ventilated first moment of the PSD (a superscript ‘v’ is used to distinguish this from the unventilated moment M1 .) In addition, lifting of air by the vertical wind component w produces supersaturation at a rate α w, where α is a function of T defined in Appendix A. The dynamics of supersaturation results from competition between these two effects.

3.2.

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Details of the model experiments

We denote our model sensitivity experiments by the bold-font characters X, Y, Z and W. The microphysical settings for the experiments are as follows. A summary is presented in Table 1. Experiment X uses the diagnostic-split PSD and a set of mass–diameter and fall-speed–diameter relations from the data collated by Mitchell (1996). The Vt –D relations for this model are shown by the solid black line (ice) and black–white dashed line (snow) in Figure 2. This configuration may be considered the ‘control’ set-up in this study. In terms of ice-cloud microphysics, the settings are those used operationally in the Met Office UKV model until January 2012. The other experiments are a sequence of changes applied to this set-up. Experiments Y, Z and W use the moment-estimation PSD parametrization. Experiment Z is obtained from X by changing the PSD type to the moment-estimation PSD. Because the moment-estimation PSD does not distinguish between different categories of ice hydrometeors, experiment Z has a single m–D relation and a single Vt –D relation. The fall-speed relation is the relation used for the snow category in the diagnosticsplit experiment X. The effect of this fall-speed change is to increase the terminal velocities of ice particles with sizes less than approximately 600 μm, relative to experiment X. The m–D relation for experiment Z is that of Cotton et al. (2012). In experiment W, the freedom to prescribe two distinct fallspeed relations for the moment-estimation PSD is re-introduced using the dual fall-speed approach described in section 3.1.3. The snow fall-speed relation is the relation used in experiments X and Z and applies in grid boxes where the PSD is relatively broad. The ice fall-speed relation (solid white line in Figure 2) has been changed to reduce the fall speeds of small crystals relative to experiment X and applies where the PSD is relatively narrow. Experiment Y is chosen to test the effects of fall-speed variability further. The fall speeds, shown by the dashed white line in Figure 2, are faster for small sizes than in the other experiments. The m–D relation is that of Brown and Francis (1995) and the moment-estimation PSD is used. 3.3.

Comparison of PSD moments

It is interesting to evaluate the different PSD parametrizations directly against the in situ aircraft measurements. Figure 3 compares two model microphysical configurations with in situ observations from the Constrain field campaign. The in situ IWCs measured with a hot-wire Nevzorov probe were used to predict the mass-carrying PSD moments for parametrizations X (black crosses) and Z (grey circles). These moments were then used with the in situ measured temperatures to predict the other PSD moments, which were compared with their measured values. Because the small particle parts of the measured PSDs are > uncertain, we compare the truncated moments M> 0 , M1 and > M3 of the measured and predicted PSDs, which include only particles larger than 100 μm. A superscript ‘>’ is used to distinguish these from complete PSD moments. The predictions are plotted against the moments measured by the aircraft cloudphysics probes. Each plotted point corresponds to an average over 10 s of flight track. The instrumentation used is discussed briefly in section 5 and in detail in Cotton et al. (2012). Figure 3 shows that the microphysical parametrization used in experiment X overpredicts the number density of particles greater than 100 μm (M> 0 ) and the truncated first moment M> . This parametrization uses the diagnostic-split PSD together 1 with the m–D relations for ice and snow given in Table 1. The parametrization used in experiment Z gives a better representation of the first two truncated PSD moments. This parametrization uses the moment-estimation PSD and the m–D relation derived from the Constrain data by Cotton et al. (2012).


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K. Furtado et al. Table 1. Models used for the high-resolution simulations. SI units.

Model

Description

References

X

Diagnostic-split PSD Mass–diameter: ice ai = 0.587, bi = 2.45 snow aa = 0.0444, ba = 2.1 Fall-speed–diameter: ice ci = 74.4, di = 0.639 snow ca = 14.3, da = 0.416 Moment-estimation PSD Mass–diameter: a = 0.0257, b = 2.0 Fall-speed–diameter: c = 14.3, d = 0.416 Moment-estimation PSD with dual fall speed Mass–diameter: a = 0.0257, b = 2.0 Fall-speed–diameter: ice ci = 1042, di = 1.0 snow ca = 14.3, da = 0.416 Moment-estimation PSD Mass–diameter: a = 0.0185, b = 1.9 Fall-speed–diameter: c = 8.20, d = 0.289

Houze et al. (1979) Mitchell (1996) Mitchell (1996)

Z

W

Y

(a)

(b)

Brown and Francis (1995)

(c)

(–)

(m–2)

(m–3)

(m–3)

Mitchell (1996) Field et al. (2007) Cotton et al. (2012)

(m–2)

(–)

3 −3 Figure 3. Comparison of PSD parametrizations with in situ aircraft data for flights b500, b503 and b497. Only points with M> (on the 2DC probe) and 0 > 10 m q > 10−6 kg kg−1 (on the Nevzorov probe) are shown. Note that the observed moments are taken from a composite PSD that is composed of data from various aircraft probes (see section 5). The superscripts ‘>’ in that notation for the PSD moments indicate that only particles greater than 100 μm in size are included.

For the higher order moments, M> 2 and greater, the two parametrizations give a similar level of agreement with the observations. This agrees with the findings of Field et al. (2007), within the temperature range of the Constrain observations. However, Field et al. (2007) also showed that the Houze et al. (1979) parametrization significantly overpredicts observations for temperatures warmer than −35◦ C.

In steady-state conditions, ρ Vm is balanced by the vertical flux of total water. In addition, we assume that the rate of increase of supersaturation due to vertical lifting is balanced by deposition of vapour on to ice and S attains the so-called quasi-steady value (Korolov and Mazin, 2003):

4. Steady states

where w is the vertical velocity, α is defined in Appendix A and τp is the phase-relaxation time-scale given by Eq. (4). The sedimentation flux can therefore be written as

If a cloud or cloud system is in a steady state, then its parameter sensitivity can be described in terms of algebraic equations. In this section we derive equations for the vertical IWC and supersaturation profiles of a steady-state ice cloud. We show that these profiles adequately describe the large-scale mean structures of the frontal system cases b500 and b503. The orographically forced case b497 is not captured by the steady-state model. We investigate the budgets of supersaturation and total water for each case and show that it is reasonable to treat the frontal systems as approximately steady-state but that the orographic case is non-steady, presumably because the time-scale for advection over the orography is short compared with the phase relaxation time-scale. 4.1.

Sqs = ατp w,

ρ Vm = ρs w 1 + Sqs .

(6)

Equations (5) and (6) depend on the specific ice mass, q, per unit mass of moist air, via ρ Vm and τp , which are determined by the microphysical parametrizations. Equations (5) and (6) can therefore be solved for q and S in terms of the saturated vapour flux ρs w and the microphysical parameters. The typical situation is that the underlying microphysical model leads to power laws for ρ Vm and τp :

ρ Vm = AqB ,

(7)

τp = Cq ,

(8)

D

An analytical steady-state model

In the remainder of this article we use ρ to denote IWC, i.e., the mass concentration of ice crystals per unit volume. Similarly, q = ρ/ρa , where ρa is the moist air density, is the specific ice water content. The ice mass sedimentation flux, ρ Vm , is the product of the IWC, ρ , and mass-weighted mean fall speed, Vm .

(5)

where the coefficients A, B, C and D are functions of the microphysical parameters a, b, etc., and may also depend on temperature. Note that Eqs (7) and (8) are not prescribed relations; they emerge from the microphysics scheme closures. Given Eqs (7) and (8), we can solve Eqs (5) and (6) by iteration


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High Cloud Sensitivity to Ice Size and Fall Speed to obtain

(a)

ρ w 1/B

α wC ρs w (D+1)/B , A B A ρ w D/B s S = α wC . A s

+

(9) IWC flux (kg m–2 s–1)

q=

(10)

Equations (9) and (10) allow the vertical profiles of q and S to be estimated if the profiles of ρs , α and w are known. If Eqs (9) and (10) can be shown to be an adequate description of simulated cloud structures, then they could provide a useful method for assessing the parametric sensitivity of numerical models. For example, given reference T, p and w profiles, Eqs (9) and (10) could be compared for a range of microphysical assumptions to estimate how simulated clouds would respond to parameter changes. Average cloud structures for the simulated cases

In this section, we compare the steady-state solutions proposed in section 4.1 with the large-scale mean vertical profiles of the three simulated cases. At individual points in the computational domains, we do not expect the model fields to correspond to local steady states. However, it is conceivable that on large scales the mean cloud structures do resemble steady states, since small-scale dynamical transients will be averaged out and the large-scale forcing of the system is relatively slow. To study the large-scale properties of the simulated cloud systems, we area-average to obtain the mean vertical structures. For each simulated case, averaging is performed within a cubical subvolume of the computational domain that encloses the majority of the cloud system. The large grey rectangles in Figure 1 show the averaging areas. A point x contributes to the average if q(x) ≥ 10−6 kg kg−1 , the cloud fraction φ (x) = 1 and no liquid water is present. We denote filtered area-averages by (·). For example, the average IWC profile of the cloud system is ρ (z). For each simulation, the model profiles of ρs w, w and α can be substituted into Eqs (9) and (10) to determine estimated steadystate profiles of q and S. The coefficients A, B, C and D can be determined for each simulation by fitting the model profiles of ρ Vm and τp to power-law functions of q. Figure 4 shows the power-law fits for case b503. The coefficients can also be found from the parametrizations, without recourse to simulated data, if T and p profiles are assumed (see Appendix B); the fits simply represent a pragmatic way of obtaining the same information. All three cases result in similar power laws and we have permuted the fit coefficients between the cases and found little qualitative effect. Figure 5 compares the steady-state estimates (circles) with the model profiles (lines) for the three cases. For the two frontal cases, the model and steady-state profiles are similar. In particular, the ordering of the analytical solutions is the same as for the model profiles. This implies that the steady states capture the parameter sensitivities seen in the model experiments. We can therefore interpret the model differences in terms of the steady-state solutions. The differences in the steady states are due to changes in the coefficients A, B, C and D. The left-hand side of Eq. (9) is dominated by the first term, so the specific ice water contents q1 and q2 in a pair of experiments are approximately related by A1 qB1 1 ≈ A2 qB2 2 . Hence if B is constant and A increases, e.g. A2 > A1 , then q2 will be smaller than q1 . Similarly an increase in B at constant A will increase q. Two effects contribute to changes in S: firstly, because D < 0, S increases when A increases (or B decreases); secondly, S changes in proportion to the τp prefactor C. Alternatively, we may note that Eq. (10) implies that S ∝ C/q|D| and therefore a change that decreases q will increase S, unless there is a compensating change in either C or D.

q (kg kg–1) (b)

tphase (s)

4.2.

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q (kg kg–1) Figure 4. Power-law fits of the model data for (a) IWC flux ρ Vm and (b) phase relaxation time-scale τp , for flight b503. The circles show the model data: filled black circles denote experiment X; open black circles experiment Z; filled white circles experiment W; and open white circles experiment Y. The straight lines show the power-law fits.

Changes to A and B are caused by changing the PSD function, m–D relation or Vt –D relation. For example, as shown in Appendix B, increasing the Vt prefactor c gives an increase in A, which leads to a reduction in steady-state IWC and an increase in supersaturation. The τp coefficients C and D change when the PSD function or m–D relation are changed. For example, a change in the PSD function that reduces the number concentration of small particles would increase the value of C. Figure 3 shows that the change from diagnostic-split to moment-estimation PSD has such an effect. Experiments X, Z and Y have different combinations of PSDs, m–D relations and Vt –D relations. The lines of IWC flux as functions of q for these experiments are roughly parallel in Figure 4, which suggests that, to first approximation, the ρ Vm exponent B can be considered the same in the three experiments. Figure 4 also shows that A is smallest in experiment X and largest in experiment Y. Hence, Eq. (9) implies that q decreases from X to Y, as seen in Figure 5. Figure 5 shows that the supersaturations are higher in experiments Z and Y than in experiment X. This is partly due to the increase in A. However, the biggest effect on S is due to the change in the τp prefactor C. Figure 4(b) shows that C is significantly lower in experiment X. This is because the diagnostic-split PSD has higher M1 for a fixed IWC, due to the larger numbers of small particles.


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Height (km)

Height (km)

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Height (km)

(c)

q (kg kg–1)

q (kg kg–1)

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(f)

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(e)

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Figure 5. Comparison of the area-averaged model profiles (lines) with the analytic steady-state solutions (circles): diagnostic-split PSD experiment (X), solid black lines and filled black circles; first single fall-speed moment-estimation PSD experiment (Z), dashed black lines and open black circles; moment-estimation PSD with dual fall-speed parametrization (W), solid white lines and filled white circles; second single fall-speed moment-estimation PSD experiment (Y), dashed white lines and open white circles. Panels (a) and (b) are for b503, panels (c) and (d) for b500 and panels (e) and (f) for b497.

There is a small increase in C between Z and Y due to the change in the m–D relation: from Eq. (10), this acts to increase the supersaturation. The increase is further enhanced by the increase in A and the result is that Y has the highest supersaturations of the four experiments. The coefficients A and B are both increased in experiment W, relative to the other experiments. Figure 4(a) shows that the net effect is that the IWC flux relation for W is similar to that for X. Consequently, we expect W and X to have similar profiles of q, as seen in Figure 5. Finally, Figure 4(b) shows that the τp relations are similar for experiments Z and W. Hence, differences in S between experiments Z and W are due to q being larger in experiment W. In particular, Eq. (10) explains why the supersaturation is lower in experiment W than in experiment Z.

4.3.

Effect of varying capacitance

The steady-state solutions (Eqs (9) and (10)) can be used to deduce the effect of varying the depositional capacitance, Cd . It might be anticipated that decreasing Cd will decrease IWC, because the growth of ice proceeds more slowly. However, the steadystate solutions depend on Cd only via the τp prefactor C. Hence, because Eq. (9) depends on C via the smaller of the two terms on the left-hand side, the profile of q will only be weakly affected by changing capacitance. The main effect of capacitance change is to change the supersaturation. Since C is inversely proportional to the capacitance, Eq. (10) implies that decreasing the capacitance will increase S. Figure 6 shows the effect of changing the capacitance on the mean profiles of q and S for case b500. The mean profiles from


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Height (km)

(a)

S (–)

Height (km)

(b)

q (kg kg–1) Figure 6. Effect of varying the capacitance for case b500. Model W (C = 1): black line and filled circles. Dashed line and open circles: effect of setting C = 1/2 in model W.

two experiments are shown together with their corresponding steady-state estimates (circles). The solid black lines and filled circles show the profiles for model W, for which Cd = 1. The dashed lines and open circles show the profiles obtained with Cd = 1/2, leaving the other microphysical parameters unchanged. Figure 6(a) shows that halving the capacitance approximately doubles S at a given height, in agreement with the steady-state predictions. Figure 6(b) shows that the IWC profile is not strongly affected. Based on Eq. (10), we would expect to see a small increase in q as a result of increasing the supersaturation. In fact, there is a small reduction in q lower down in the cloud. This is due to changes in the saturated part, ρs w, of the vertical moisture flux. In general, any two experiments can have different saturated vapour fluxes, due to feedback of the microphysical changes on the temperature structure. We have found this effect to be small for the experiments performed and the parameters A, B, C and D determine the changes to the q and S profiles. However, Eqs (9) and (10) imply that, if two experiments have very similar microphysical parameter values, then changes in the cloud structure due to differences in ρs w may be more obvious. For example, when only the capacitance is varied there are no changes to the coefficients A and B, i.e. the sedimentation flux is unchanged for a fixed IWC. Consequently, the effect on q of differences in ρs w can be noticeable. 4.4.

Steady-state budgets

In section 4.2, we compared the analytic steady-state solutions with the model profiles for the simulated cases. The reasonable level of agreement suggests that steady-state assumptions can

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be useful for understanding model clouds. The steady-state solutions used were derived from Eqs (5) and (6). We can further assess the validity of the steady-state assumptions by directly considering how well Eqs (5) and (6) hold for each simulation. Complementary information on the degree of non-steadiness can be provided by analysing the mean budgets for supersaturation and total water transport. Using a representative simulation as an example, we show that the large-scale steady state approximation is reasonable for the two frontal cases, although it becomes less applicable close to the cloud top. In Figure 7, we examine the extent to which a large-scale steady state emerges for experiment Z for case b500. In Figure 7(a), the area-averaged vertical supersaturation profile is shown by the solid line. The dashed line shows the mean vertical profile of the quasi-steady supersaturation, Sqs = ατp w. The agreement shows that the quasi-steady approximation captures the vertical variability of the mean supersaturation over a large cloudy area. It is, however, an underestimate and the approximation becomes quite poor very close to the cloud top. Quasi-steady supersaturation results because supersaturation production and depletion are approximately balanced, as shown in Figure 7(b). The solid line shows the profile of the mean rate of vapour depletion due to deposition, S/τp . The dashed line shows the mean rate of production due to lifting, α w, which is seen to approximate the solid line. The discrepancy between the two terms is mainly due to advection of moisture and temperature. These effects can be seen from the dot–dashed line, which shows the combined tendency due to production and advection, −w · ∇ S, of supersaturation. In Figure 7(c), we consider the components of the vertical total water flux. The solid line shows ρ Vm , the mean ice sedimentation flux. The dashed line shows (ρv + ρ )w, the mean total water flux due to vertical advection, which is in close agreement with the sedimentation flux. The steady-state solutions are based on using Eq. (6) for total water transport. The dot–dashed line in Figure 7(c) shows the estimated flux ρs w(1 + Sqs ), which corresponds to the left-hand side of Eq. (6). The steady-state moisture flux is seen to provide a reasonable estimate of the sedimentation flux. In fact, it provides an underestimate because we have neglected the contribution ρ w. Figure 7(c) indicates that total water transport is a balance between upward transport of water vapour and downward sedimentation of ice particles. This conclusion is borne out by Figure 7(d), which shows the contributions to the total water budget. The solid line shows the absolute value of the ice sedimentation rate, −∂z ρ Vm /ρa . The dot–dashed line shows the total water advection increment, −w · ∇ (qv + q), where qv is the specific humidity. The dashed line shows the total water increment, −w∂z (qv + q), due to vertical advection only, the greater part of which is vertical transport of water vapour (not shown). Figure 7(d) suggests that vertical large-scale total water transport is in an approximate steady state. Similar conclusions hold for all the experiments performed for the two frontal cloud cases: we find that the large-scale steadystate approximation is reasonable but is least good above 10 km (indicating a degree of non-steadiness in the uppermost parts of the model clouds). In contrast, we find that the same is not true for the orographically forced case b497: the advection of total water is still balanced by the ice crystal sedimentation rate but a large part of this is horizontal advection. 5. Comparisons with observations In this section, the results of the model sensitivity experiments are compared with in situ aircraft measurements. The in situ observations used here were from the FAAM BAe-146 aircraft. IWC was measured with a hot-wire deep cone Nevzorov probe (Korolov et al., 1998), shown by Cotton et al. (2012) to be sensitive to IWCs as low as 0.001 g m−3 . A General Eastern chilled


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Height (km)

(b)

Height (km)

(a)

Tendency (s–1)

S (–)

Height (km)

(d)

Height (km)

(c)

IWC flux (kg m–2 s–1)

Tendency (s–1)

Figure 7. Budgets for moisture and ice mass transport in the simulation of experiment Z for case b500. The model fields have been averaged over the large grey area shown in Figure 1(a). (a) Mean humidity structure: mean supersaturation profile, S, solid line; mean profile of the quasi-steady supersaturation, Sqs , dashed line. (b) Supersaturation budget: mean rate of depletion of S due to deposition, solid line; mean rate of production S due to vertical lifting, dashed line; combined tendency due to advection and production by lifting, dot–dashed line. (c) Comparison of vertical fluxes: ice particle sedimentation flux, solid line; average vertical flux of total water, dashed line; estimated IWC flux given by Eq. (6), dot–dashed line (assumes quasi-steady supersaturation and neglects IWC transport). (d) The average total water budget: sedimentation rate of ice crystals, solid line; averaged vertical advection of specific total water mass, dashed line; averaged total advection (including horizontal component) of specific total water mass, dot–dashed line.

mirror hygrometer was used to provide humidity information. Ice particle sizes were measured with a Particle Measuring Systems 2D-cloud (2D-C) probe, a SPEC Incorporate 2D-S probe and a Droplet Measurement Technologies Cloud Imaging Probe (CIP100). The 2D-C and CIP-100 were fitted with modified tips (Korolov et al., 2011) to reduce contamination of the measured size distributions by particles shattered on the probe housings. The 2D-S did not have modified tips. However, for the 2D-C and 2D-S, is was possible to apply inter-arrival time filtering to reduce the effects of shatter artefacts further. In this article, we use a composite size distribution, which was derived by combining data from all the probes after inter-arrival time filtering had been applied. Full details of the data analysis can be found in Cotton et al. (2012). For comparison with the models, only data for particle sizes greater than 100 μm was used. Because most of the diagnosed shattering occurred on the 2D-S probe, it is also convenient to compare the model fields directly with the 2D-C measurements, as well as with the composite PSD. A consistent criterion was used to define when data (aircraft or model) constituted ‘in-cloud’ data. A 10 s interval of aircraft measurements was regarded as being in-cloud if the IWC measured by the Nevzorov probe exceeded 10−6 kg m−3 and the number concentration of particles greater than 100 μm, measured

by the 2D-C, exceeded one particle per litre. To compute the incloud statistics for the model fields, points were filtered according to the same criteria: hence a model grid box, x, contributed to the 3 −3 in-cloud statistics if ρ (x) > 10−6 kg m−3 and M> 0 (x) > 10 m . 5.1.

Comparisons with in situ observations

For comparison with the in situ data, the instantaneous model fields were averaged over the 104 km2 areas shown by the small grey squares in Figure 1. For each case, the time of the comparison falls within the duration of the sortie. The areas and times were chosen based on (i) proximity to the aircraft operating region and (ii) the degree of agreement between the model and observed pressure, potential temperature and horizontal wind component profiles. In Figures 8, 9 and 10 model fields are compared with aircraft observations from each of the three cases. The thick lines correspond to the model sensitivity tests performed: the solid black line is the diagnostic-split PSD experiment (X); the dashed black line is the first single fall-speed, momentestimation PSD experiment (Z); the solid white line is the dual fall-speed, moment-estimation PSD experiment (W); the dashed white line is the second single fall-speed, moment-estimation PSD experiment (Y).


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RHI (–) (d)

(f)

Height (km)

(b)

M2 (m–1)

q (kg kg–1)

M1 (m–2)

M3 (–)

Figure 8. Comparison of aircraft in situ observations with the model profiles for the frontal cirrus case b500. (a) Relative humidity with respect to ice; (b) in-cloud > specific ice water content, q; (c)–(e) successive truncated moments of the ice PSD from M> 0 to M3 , including only particle sizes greater than 100 μm. The circles show the observation means for each straight-and-level run. Open and filled circles correspond to successive spiral descents. The horizontal variation bars cover two standard deviations of the data at each height. The curves show the mean model profiles in the selected averaging region: experiment X, solid black line; Z, dashed black; W, solid white; Y, dashed white.

(c)

(e)

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(a)

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RHI (–) (d)

(f)

Height (km)

(b)

M2 (m–1)

q (kg kg–1)

M1 (m–2)

M3 (–)

Figure 9. As Figure 8, but for the frontal cirrus case b503.

The circles show the mean in situ measurements for each straight-and-level run. The open and filled circles correspond to the first and second spiral descents performed on each day. The horizontal variation bars cover ±1 standard deviation around the means. In Figures 8(a), 9(a) and 10(a), the model relative humidity (RHI) profiles are compared with those measured by the aircraft hygrometer (circles) and by dropsonde (dotted black lines). In Figures 8(b), 9(b) and 10(b), the circles show the mean measured in-cloud specific ice water content. Note that where no data was found to satisfy the imposed criteria for an in-cloud sample, the observational mean appears as zero. This was a common

occurrence on the first attempted descents for the two frontal cases. Panels (c)–(f) of Figures 8, 9 and 10 show the in-cloud values of the first four size-distribution moments for particle sizes greater than 100 μm. The circles show the in-cloud means calculated from the composite PSDs and therefore include data from all the available probes. The crosses (times signs, ×) show the mean in-cloud values obtained from the 2D-C probe. Due to variations in probe sample volumes and data processing, these are typically lower than the composite PSD values. The thin horizontal lines link the two means for each run.


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RHI (–) (d)

(f)

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(b)

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q (kg kg–1)

M1 (m–2)

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Figure 10. As Figure 8, but for the orographic cirrus case b497.

Panels (a) and (b) of Figures 8–10 confirm the conclusions of section 4.2 regarding the sensitivity of cloud-top RHI and q to the parametrization changes. The diagnostic-split PSD typically gives much drier cloud tops than the moment-estimation PSD experiments. Amongst the moment-estimation PSD experiments, we see that increasing the fall speed of the small ice crystals results in a decrease in cloud-top IWC and an increase in cloud-top RHI. In section 4.2, these effects were explained in terms of a shift in the model equilibrium state towards lower IWC in response to microphysical changes, which increased the mass sedimentation flux of ice at a fixed value of q. From Figures 8–10, it is evident that similar conclusions also hold for smaller averaging areas. This suggests that small-area averages continue to exhibit some of the character of larger scale mean states, presumably on account of being drawn from an ensemble of possible states centred around the large-scale mean. In terms of the general agreement between observations and model, panels (c) and (d) of Figures 8–10 show the diagnosticsplit PSD overpredicting the number concentration and first moment of the PSD for particles greater than 100 μm. Case b500 suggests that the increased cloud-top humidity due to the moment-estimation PSD is an improvement over the diagnostic PSD model. However, for cases b503 and b497 the cloudtop air is close to ice-saturation and the diagnostic-split PSD gives the best RHI prediction. Ice-saturated cloud tops with the diagnostic-split PSD result from the overestimated first moment, suggesting that the humidity structures for experiment X result from misrepresenting the microphysical state of the clouds. The in-cloud IWC is closest to the observations for the diagnostic-split PSD experiment X and the dual fall-speed moment-estimation PSD experiment W. The moment-estimation PSD experiments that use a single Vt –D relation underestimate the in-cloud IWC close to the cloud top, because the fall speeds are too large for small particle sizes. The freedom to model the fall speeds of small and large particles independently is seen to allow more realistic cloud-top ice masses to be obtained. The effects of having too high an ice fall speed are most noticeable for experiment Y. For all cases, this experiment significantly underpredicts the ice water path of the cloud and overpredicts the relative humidity. Together with experiment Z, this highlights the importance of reconsidering the choice of fall speeds in conjunction with changes to ice PSDs.

Table 2. Control model parameters for the global climate simulations. SI units. Model

Description

References

H

Diagnostic-split PSD Mass–diameter: ai = aa = 0.0185, bi = ba = 1.9 Fall-speed–diameter: ice ci = 201, di = 0.679 snow ca = 8.20, da = 0.289

Houze et al. (1979) Brown and Francis (1995)

6. Climate simulations In this section, we consider the effects of changing ice fall speed and PSD on a 20 year climate simulation. The control model, H, uses the Met Office Global Atmosphere (GA5) configuration (Walters et al., 2013). Table 2 specifies the ice microphysics for this model. The experiment is based on the GA5 configuration, but the ice microphysical settings are changed to be the same as the high-resolution experiment W. Any differences in climate between models H and W are therefore due to a combination of PSD change, mass–diameter change and a reduction in particle fall speed of approximately a factor of two. Figure 11 compares the ice Vt –D relations of the two models: the white lines show the relations for model W; the black lines those for model H. The ice cloud optical properties for both simulations use the effective diameter scheme of Edwards et al. (2007). Figure 12 shows the zonal-mean IWC, specific humidity, temperature and relative humidity in the experiment minus those in the control. The means shown include only the Northern Hemisphere winter periods (December–February). Figure 12(a) shows that the IWC is lower in the control throughout the troposphere. This is expected, based on the results of section 4, if the microphysical differences are such that the IWC flux, ρ Vm , at fixed values of q is lower in experiment W. Figure 11 suggests that this is indeed the case due to the changes in fall speed. Figure 12(b) shows that the specific humidity has also increased in the experiment. The temperature changes are shown in Figure 12(c). The increase in IWC is accompanied by a warming of the troposphere and simultaneous cooling of high cloud regions and


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Vt (m s–1)

of changing from the diagnostic-split to the moment-estimation PSD. Alternatively, this can be verified directly from the data given in Tables 1 and 2. 7. Conclusions

diameter (µm) Figure 11. Fall-speed–diameter relations for the two climate simulations. For control model H: dashed black line, snow fall speeds; solid black line, ice fall speeds. For the experiment climate simulation using the moment-estimation PSD experiment with dual fall speeds: dashed white, snow fall speed; solid white line, ice fall speed. Thin grey lines show the data from Mitchell (1996).

the lower stratosphere due to changes to the radiative heating rates. Figure 12(d) shows the changes in RHI that result from the increased atmospheric water vapour and temperature changes. The RHI of the upper troposphere has increased, particularly at heights close to the climatological cloud tops. We expect model W to have higher cloud-top RHI if it exhibits longer phase relaxation times at fixed q than the control model. The results of the high-resolution case studies show that τp increases as a result

We have investigated the sensitivity of high cloud simulated by an operational weather forecast model to ice microphysical parameters. This has been done in the context of high-resolution case studies of midlatitude cirrus and global climate simulations. The high-resolution sensitivity tests were evaluated against aircraft observations. We have shown that the sensitivity of the models studied can be understood in terms of steady-state solutions to the model equations. These solutions depend explicitly on the parameters in the cloud microphysics scheme and hence their response to parameter changes can be deduced by inspecting formulae without having to resort to resource-intensive numerical experiments. The steady-state methodology is applicable to other numerical models and could be used to interpret the results thereof. For frontal cloud systems, we have shown that the response of IWC and RHI to parameter changes can be considered as due to the requirement that, on average, (i) the downward flux of ice mass must balance the upward flux of water vapour due to largescale ascent and (ii) the rate of production of supersaturation due to lifting is balanced by the rate of depletion due to deposition on to ice. From these requirements, we can draw the following conclusions connecting macrophysical properties, i.e. RHI and

pressure (mb)

(a)

(

)

pressure (mb)

(b)

( )

pressure (mb)

(c)

( )

pressure (mb)

(d)

Figure 12. Model field differences between climate model W and the control model H: (a) ice water content, (b) specific humidity, (c) temperature and (d) relative humidity.


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IWC, of frontal high clouds to the microphysical parameters that affect them. (1) The IWC at a given height can be increased by changes that decrease the sedimentation flux, ρ Vm , for a fixed IWC. Microphysically, such changes can be brought about by adjusting the fall-speed–diameter relation or particle size distribution. (2) The relative humidity at a given height can be increased by changes that increase the phase relaxation time-scale for a fixed IWC. Such changes will result from decreasing the ventilated first moment, Mv1 , of the PSD or from decreasing the depositional capacitance, Cd . It is noteworthy that the depositional capacitance has a significant effect on RHI only, whereas its impact on IWC is very weak, a result that is perhaps counter-intuitive unless explicit steady-state formulae are used. We have evaluated the model IWCs, size distribution moments and relative humidities against in situ measurements from the Constrain field campaign (Cotton et al., 2012). From this we draw the following conclusions. (1) The experiment with the diagnostic-split PSD based on Houze et al. (1979) overestimated the low-order moments of the size distribution. The experiments performed with the moment-estimation PSD from Field et al. (2007) gave a better representation of the in situ PSD moments. (2) The large values of M1 for the diagnostic-split PSD lead to the in-cloud air being close to ice saturation. For the moment-estimation PSD experiments, the in-cloud air had greater RHI because of the larger values of τp . More case studies would be useful for constraining the model cloud-top relative humidity. (3) The experiment with the diagnostic-split PSD gave reasonable predictions of the observed IWCs. When using the moment-estimation PSD, a hybrid fall-speed relation formed from two power laws was needed to reproduce the observed IWCs. We compared climate simulations for two different microphysics configurations. A simulation performed with the Met Office GA5 configuration provided a control simulation, against which the effect of simultaneously changing the ice PSD, fall speed and mass–diameter was tested. This combination of changes caused the phase relaxation time-scale to increase and the IWC flux to decrease, for a fixed IWC. In correspondence with the highresolution simulations, these changes brought about increases in both upper tropospheric RHI and IWC.

B0 = 2π

−1 ,

(A1)

where ρa and κ are the density and thermal conductivity of air, es is the saturated water-vapour pressure over ice and Dv is the molecular diffusion coefficient of water vapour in air. The ventilation factor is Fv = P + QSc1/3 Re1/2 , where Re = Vt D/ν is the particle Reynolds number, Sc is the Schmitt number (assumed constant) and P and Q are constants. The bulk growth rate of the particle population is found by ˙ over the PSD. Combining the transport equations integrating m for moisture and temperature gives Squires’ equation for the

(A2)

where τp = 1/Cd β B0 Mv1 is the phase relaxation time-scale, the ventilated first moment of the PSD is c 1/3 M(3+d)/2 v M1 M1 = P + Q Sc (A3) ν M1 and the coefficients β and α are given by L2S 1 + , ms cp Rv Tv2 gLS g α= − . cp Rv T 2 Ra T

β=

(A4) (A5)

In principle, the denominator of the first term in Eq. (A4) should contain the mixing ratio of water vapour, in place of the saturated value ms , but this does not appreciably alter Eq. (A2). The quasi-steady supersaturation is the solution of Eq. (A2) obtained when the Lagrangian time derivative of S is zero. For small enough values of S, the Eulerian steady-state supersaturation satisfies S = Sqs − τp w · ∇ S, which will be similar to Sqs provided the advection tendency −w · ∇ S is small. Appendix B: Parameter dependencies The coefficients A, B, C and D in the power laws that relate the IWC flux, ρ Vm , and phase relaxation time-scale, τp , to the specific IWC, q, are determined by the PSD parametrization, m–D and Vt –D relations. Here, as an illustration, we derive expressions for the prefactors A, C and exponents B, D in the ρ Vm and τp power laws for an exponential PSD. Consider an exponential PSD of the form f (D) = N0 (T) exp(−λD), where the intercept parameter N0 (T) is a prescribed function of temperature and λ is obtained from the standard single-moment closure: λ = (aN0 (δ )/ρ )1/δ , where δ = b + 1 and is the gamma function. The sedimentation flux is given by the (b + d)th moment of the PSD. For small ice crystals, τp is dominated by the first moment of PSD. Hence, for an exponential PSD, c (κ ) 1−γ 1−γ γ N a ρ ,

(δ )γ 0

(δ )μ μ−1 μ −μ τp = N a ρ , Cd β B0 0

ρ Vm =

(B1) (B2)

where γ = 1 + d/δ , κ = γ δ and μ = 2/δ . From Eqs (B1) and (B2) we can make the following identifications: c (κ )ρa 1−γ 1−γ N0 a , (B3)

(δ )γ

(δ )μ ρa−μ μ−1 μ N0 a , (B4) C= Cd β B0 d , (B5) B=γ =1+ b+1 2 (B6) D = −μ = − . δ The following observations can be made regarding the effects of parametrization changes on ρ Vm and τp at constant values of ρ . Firstly, the sedimentation flux scales linearly with the Vt prefactor, c, and τp is inversely proportional to the capacitance, Cd . Since γ > 1 and, typically, δ > 2, a change that increases the intercept N0 will decrease the sedimentation flux and phase relaxation time. Similarly, a change that increases the m–D prefactor, a, will decrease the sedimentation flux but increase the phase relaxation time. The sensitivity to parameter changes involving the exponents b and d is somewhat more involved and will not be described here. A=

We use the standard parametrization for crystal growth, whereby the rate of change of mass of an individual crystal is m ˙ = Cd B0 DFv S, where Cd is the depositional capacitance, Fv is a factor accounting for ventilation effects and B0 is an effective water-vapour diffusivity given by

ρa L2S ρa Rv T + 2 κ Rv T es Dv

(1 + S)S dS =− + (1 + S)α w, dt τp

γ

Appendix A: Supersaturation dynamics

evolution of the supersaturation S:


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Q. J. R. Meteorol. Soc. 141: 1546–1559 (2015)