Review of moist processes

We have carried out a recent review of the role of moist convection in hurricanes and its representation in hurricane models. Further details may be found in Smith (2000). The release of latent heat in deep convection leads either directly or indirectly to buoyancy forces that contribute to driving the secondary circulation of the vortex (i.e. the circulation in a vertical plane). The way in which deep convection generates buoyancy is discussed in the foregoing paper. The definition of buoyancy in rapidly rotating vortices is considered in a paper by Smith and Zhu (2001).
 

A minimal three-dimensional tropical-cyclone model

1. Introduction

A minimal three-dimensional numerical model designed for basic studies of tropical cyclone behaviour has been developed (Zhu et al., 2001a, 2001b). The model is formulated in s-coordinates and can be run on an f- or b-plane. It has three vertical levels, one characterizing a shallow boundary layer and the other two representing the upper and lower troposphere, respectively. It has three options for treating cumulus convection on the subgrid scale and a simple scheme for representing the explicit release of latent heat on the grid scale. Here we describe briefly and compare these schemes, which are based on the mass-flux models suggested by Arakawa and Ooyama in the late 60s, but modified to include the effects of precipitation-cooled downdraughts. They differ from one another in the closure that determines the cloud-base mass flux. One closure is based on the assumption of boundary-layer quasi-equilibrium proposed by Raymond and Emanuel.

2. Description of the model

The model is based on the three-dimensional hydrostatic primitive equations in sigma coordinates (x, y, s) on an f-plane, where x and y are in the zonal and meridional directions, respectively, and s = (p– p_top) /(p_s – p_top ) ,  p* = p_s – p_top , p_s and p_top are the surface and top pressures and  ptop is taken to be 100 mb. The configuration of the staggered vertical grid is shown in Fig. 1.

Fig. 1.  Configuration of s–levels in the model showing locations where the dependent variables are stored. The horizontal velocity components (u_n, v_n ), geopotential (F_n), potential temperature (q_n), specific humidity (q_n), and the moist static energy (H_n) used in the convection scheme are calculated at levels n ( = 1, 3 and  b) and the vertical velocity and convective mass flux M_n are stored at levels 2 and 4. A few features of the model are:

• A Newtonian cooling term with a radiative time scale of 12 h is added.
• The turbulent flux of momentum to the sea surface and the fluxes of sensible heat and water vapour from the surface are represented by bulk aerodynamic formulae.
• The initial axisymmetric vortex is barotropic and has the same tangential velocity distribution as that used by Smith et al. (1990).
• If super-saturation occurs in a grid box at some time step, the excess water is removed as precipitation and the equivalent amount of latent heat is released. This procedure requires iteration.

3. Cumulus parameterization scheme

A .  The cloud model

A schematic of the cloud model is shown in Fig. 2.

Fig. 2: Schematic diagram of the vertical circulation within a grid box in the cumulus parameterization schemes used in this paper. The schemes are based on a steady state bulk cloud model which is assumed to occupy a small fraction m of the grid box. The cloud takes mass from the boundary layer at a rate M_c4 and detrains it into the upper layer at a rate M_c2. Entrainment of middle-layer air into the cloud takes place along the way at a rate M_e, just sufficient to ensure that the air detrains from the cloud at the same temperature as the environment at level-1 (i.e. detrainment of cloud air occurs at its level of neutral buoyancy). The downdraught mass flux M_d4 is assumed to be proportional to M_c4 and the downdraught is considered to be a part of the environment. The resolved-scale flow in or out of the grid box (indicated in the diagram by arrows through the right boundary) has an associated mass flux   (n = 2, 4). If  is less than the net convective mass flux at a particular level, there is subsidence in the cloud environment which warms and dries the environment in the layer below. The difference between the three schemes lies entirely in the method of closure that determines M_c4.

B.  Entrainment
 

An energy budget determines the rate of mass entrainment, M_e, into the model cloud:

M_e = (h - 1)M_c4 ,    where  ,

where H_n is the moist static energy at level-n and H_1s is the saturated moist static energy in the upper layer

C.  Mass flux above the boundary layer

(i)  Modified 1969 Arakawa scheme

M_c4  is obtained by relaxing the vertical distribution of moist static energy in a grid column to a moist adiabat, i. e.

.

(ii)  Modified 1995 Emanuel scheme

M_c4 is obtained by relaxing towards the mass flux M_eq determined on the assumption that the boundary-layer is in quasi-equilibrium.

Fig. 3. Schematic illustration of the budget of moist static energy that determines the equilibrium mass flux in the Emanuel closure.

(iii)  Modified 1969 Ooyama scheme

M_c4  is obtained by assuming that M_c4- M_d4 equals the resolved mass convergence rate in the boundary layer.

4. The numerical experiments
 

The model is initialized with an axisymmetric vortex in gradient wind balance (with maximum tangential wind speed v_max = 15 m/sec at a radius of r = 120 km).

Four calculations are carried out:

• Explicit moist processes only;
• One of three subgrid-scale deep convection schemes:

-  Arakawa closure

-  Emanuel closure

-  Ooyama closure


The intensification rate for the four experiments is shown in Fig. 4.

Maximum surface wind speed (m/sec)



Fig. 4. Variation of maximum surface wind speed in ms^-1 as a function of time in the four calculations. Numbers on curves refer to: 1  Explicit moist processes only, 2 Arakawa closure, 3 Emanuel closure, and 4 Ooyama closure.
 

5. Summary
 

• All the calculations produce a reasonably realistic model tropical cyclone including:
- the development of a warm core
- a period of gestation
- a period of rapid intensification
 
• In all cases the period of rapid intensification coincides with saturation on the grid scale in the lower troposphere.
• As a storm intensifies, the latent heat released by the parameterized convection decreases and at the mature stage the latent heat release is mainly grid-scale (explicit).
• The implementation of a convective parameterization leads to intensification before saturation occurs on the grid-scale.
• Deep convection dries the troposphere and the boundary layer => grid-scale saturation is delayed => the period of rapid intensification is delayed.
• The calculations using the Arakawa scheme and the Emanuel scheme are similar. With these the cloud base mass flux is independent of the resolved-scale boundary-layer convergence.
• The period of rapid development using the Ooyama scheme occurs earlier than with the other schemes. The convective mass flux is much larger and grid-scale saturation occurs earlier, but where air columns are conditionally stable!

6. Hypothesis
 

• The period of rapid development in tropical cyclones is accompanied by a change in the character of deep convection from buoyantly-driven, predominantly upright convection to forced moist ascent in clouds.
• The change occurs when the mass convergence in the boundary-layer exceeds the cloud base mass flux.
• This is presumably what previous authors mean by “the organization of convection by rotation”?

7. Additional experiments exploring three important physics processes

Additional experiments have been carried out to explore the roles of shallow convection, precipitation-cooled downdrafts and the vertical transport of momentum by deep convection on tropical cyclone evolution (see Zhu and Smith, 2001b).

7.1   Experiments with shallow convection

Shallow convection in the tropical atmosphere plays an important role in bringing dry air (with low entropy and low moist static energy) from aloft to the  boundary layer and in transporting moist boundary layer air (with high entropy and high moist static energy) to the lower troposphere (see e.g. Betts, 1975). It is represented in the model using the method suggested by Arakawa (1969), which allows deep clouds and shallow clouds to coexist. Whether or not a grid column is stable to deep convection, it is considered to be unstable to shallow convection if h_b > h₃*. In this case, a convective mass flux M_sc4 carries boundary-layer air into the middle layer and compensating subsidence in clear air carries lower tropospheric air into the boundary layer at the same rate.

The closure assumption that determines M_sc4 is obtained in a similar way to Arakawa's method for deep convection. It is
assumed that shallow convection reduces the instability on a specified time scale, t_sc, and drives the saturation moist static
energy of the middle layer towards the moist static energy of the boundary layer.

Fig. 5 shows time-series of the maximum tangential wind speed in the boundary layer in the calculation with the Arakawa closure for values of t_sc = 2 h, 4 h, 6 h, and 10 h  (the curve for t_sc = 8 h is not included as it is found to be very close to that for t_sc = 6 h) and in the control calculation. As shown below, these values span a range over which the effects of shallow convection are exaggerated (i.e. t_sc = 2 h) to one in which they are small. It can be seen that shallow convection reduces the rate of vortex intensification during the gestation period for all values of t_sc, but the reduction is larger as t_sc decreases. However, the length of the gestation period depends on t_sc in a less uniform manner. When t_sc = 2 h, rapid intensification occurs about 15 h earlier than in the control calculation, but as t_sc increases, so does the gestation period until t_sc reaches a value of somewhere between 6 and 8 h, when rapid intensification occurs about 30 h later than in the control calculation. As t_sc increases further, the gestation period declines towards that of the control calculation; for  example, when t_sc = 10 h, the delay is only 20 h.

Fig 5. Evolution of maximum tangential wind speed in the boundary layer in the calculation with the Arakawa closure with  different adjustment time scales for shallow convection, t_sc. Numbers on the curves refer to: (0) the control calculation; (1) t_sc = 2 h; (2) t_sc = 4 h; (3)  t_sc = 6 h; and  t_sc = 10 h.

Calculations were carried out using either the modified Arakawa or Ooyama closures for deep convection. The inclusion of shallow convection leads to several competing effects on vortex evolution, the net effect being dependent on the assumed time scale for this process. These effects are listed below:
 

• The direct effect of shallow convection is to moisten and cool the lower troposphere and to warm and dry the boundary layer. In particular, the moist static energy of the boundary layer is reduced, as is the degree of instability between the two layers.
• Shallow convection diminishes the instability to deep convection, thereby reducing the deep convective mass flux, and hence the rates of deep convective heating and rate of buoyancy generation.
• The lower buoyancy associated with the cooling by shallow convection and reduced deep convective heating leads to a weaker secondary circulation and a reduced rate of vortex intensification.
•  Unless the direct moistening is very strong, the weaker secondary circulation delays saturation on the grid-scale and prolongs the gestation period.
• The elevation of the moist static energy of the lower troposphere reduces the impact of subsidence in lowering that of the boundary layer, so that surface fluxes become more effective in elevating the boundary layer moist static energy.
• The drying of the boundary layer reduces the tendency of the boundary layer to saturate as happens in the control calculation,  making the model more realistic.
• A convective time scale for shallow convection on the order of 6 h produces the most realistic distributions of boundary layer moisture and temperature in the present model.

7.2   Experiments without precipitation-cooled downdrafts convection

When the lower troposphere is relative dry, downdrafts associated with evaporating precipitation from deep convection moisten and cool the boundary layer (Betts, 1976). Two processes are involved, evaporation and downward transport. Evaporation of falling precipitation into the unsaturated sub-cloud layer is a heat sink and moisture source and brings the layer closer to saturation at constant equivalent potential temperature (or moist static energy). The cooling produces negative buoyancy, which together with the drag of falling precipitation generates the downdraft. The downdraft transports potentially warmer and drier air into the sub-cloud layer. The two processes oppose each other in the sense that evaporation cools and moistens while downward transport warms and dries. Betts points out any combination can result, but the observations he reports suggest that the sub-cloud layer becomes cooler and drier after the precipitation and downdrafts. The coolness implies that there is sufficient evaporation into subsiding air to offset the adiabatic warming and the dryness is a
consequence of the fact that even the saturation mixing ratio of subsiding air is still lower than that of the boundary layer.

Fig 6. shows time series of minimum surface pressure in the control calculations and in experiments with the three different closures in which downdrafts are excluded. It can be seen that the effects of downdrafts depend on the closure scheme used for deep convection. For the Arakawa and Emanuel closures, the rate of deepening during the early gestation period is larger when downdrafts are not included, but the onset of rapid vortex intensification is delayed by about 60 h in each case. A similar delay occurs also for the Ooyama scheme (about 40 h), but the exclusion of downdrafts leads to a smaller rate of intensification.

Fig 6  Evolution of maximum tangential wind speed in the boundary layer [in m/s] in (a) the control calculations, and (b)
in Expts. without precipitating downdrafts. Numbers on the curves refer to: (1) the Arakawa closure ; (2) the Emanuel closure; (3) the Ooyama closure.

The effects of precipitation-cooled downdrafts depend on the closure scheme used for deep convection. For the Arakawa and Emanuel closures, the rate of deepening during the early gestation period is larger when downdrafts are not included. This behaviour is as expected and may be attributed to the fact that, in our model, these downdrafts carry air into the boundary layer that has a lower moist static energy than air that subsides elsewhere. However, downdrafts delay the time of rapid vortex intensification by about two and a half days in each case, a result that would not have been anticipated. Apparently, downdrafts allow a steady build up of the deep convective mass flux and hence the secondary circulation, whereas in the case where downdrafts are excluded, the more rapid initial build up is arrested and temporarily reversed by a subsequent stabilization to deep convection. As a result the deep convective mass flux and hence the secondary circulation in the case with downdrafts eventually exceed those in the calculation without downdrafts and saturation grid-scale saturation is achieved earlier in this case. A similar delay in rapid intensification occurs also for the Ooyama scheme (about a day and a half), but the exclusion of downdrafts in this case leads directly to a smaller deep convective mass flux and therefore to a slower intensification rate.
 

7.3   Experiments with vertical momentum transfer

Deep convective clouds transport not only heat and moisture vertically, but also horizontal momentum.   To explore this
possibility we carried out experiments in which the mass fluxes associated with deep convection calculated using the Arakawa scheme transport momentum vertically between layers. The formulation is essentially the same as that for moist static energy . Momentum is transferred directly in deep convective cloud from the boundary layer to the upper layer at a rateM_c4(u_b, v_b) and from the middle layer to the upper layer at a rate M_e(u_3, v_3). At the same time momentum is transferred from the upper layer to the middle layer at the rate M_c2 (u_2, v_2)  and from the middle layer to the lower layer a the rate M_c4(u_4, v_4), where u_2, v_2 are the velocity components at the interface level-2 and u_4, v_4 are the corresponding components at level-4.

The evolution of the maximum tangential wind speed in the boundary layer in Expt. with moment transfer is compared with that in the control calculation in Fig 7. It is clear that momentum transport under these circumstances has a dramatic effect on vortex evolution by suppressing vortex development for several days; after a short period of intensification the vortex stagnates and not until about 80 h does it begin to slowly intensify. Rapid intensification occurs eventually after 150 h.

Fig 7.  Evolution of maximum tangential wind speed in the boundary layer in ms^-1 in the control calculation (curve labeled 1), and the experiment with vertical momentum transfer for the Arakawa closure.

Convective momentum transport as represented in the model weakens the secondary circulation in the vortex, which impedes vortex development and significantly prolongs the gestation period. However it does not significantly reduce the maximum intensity attained after the period of rapid development. The results are compared with those found in other studies.
 

8. References

Arakawa, A., 1969, Parameterization of cumulus convection. Proc. WMO/IUGG Symp. Numerical Weather Prediction, Tokyo, 26 Nov. - 4 Dec. 1968, Japan Meteor. Agency IV, 8, 1-6.

Betts, A. K.,  1976, The thermodynamic transformation of the tropical subcloud layer by precipitation and downdrafts.
J. Atmos. Sci.,  33, 1008-1020.

Emanuel, K. A., 1995, The behaviour of a simple hurricane model using a convective scheme based on subcloud-layer
entropy equilibrium. J. Atmos. Sci., 52, 3960-3968.

Raymond, D. J., 1995, Regulation of moist convection over the West Pacific warm pool. J. Atmos. Sci., 52, 3945-3959.

Smith, R. K., W. Ulrich and G. Dietachmayer, 1990, A numerical study of tropical cyclone motion using a barotropic model. Part I. The role of vortex asymmetries. Quart. J. Roy. Meteor. Soc., 116, 337-362.

Smith, R. K., 2000, The role of cumulus convection in hurricanes and its representation in hurricane models. Rev. Geophys., 38, 465-489.

Smith, R. K., M. T. Montgomery, and H. Zhu, 2002,  Buoyancy and the baroclinicity vector in tropical-cyclone vortices. Submitted to  J. Atmos. Sci. (Nov. 2002)

Zhu, H., R. K. Smith and W. Ulrich, 2001, A minimal three-dimensional tropical cyclone model. J. Atmos. Sci.,58, 1924-1944.

Zhu, H., and R. K. Smith, 2001, Three important physical processes in a minimal three-dimensional tropical cyclone model. J. Atmos. Sci. 59,  1825-1840.
 

Date 18 November 2002