Descriptions of Cumulus convection parameterization

          (Click to see how the equations are calculated in codes!)

  Main subroutine of cumulus parameterization, the subroutine calculates several thermodynamic terms (e.g. moist static energy at each layer) which are needed in the subroutines afterward.

  The mass flux and detrainment of cloud is,

      

      

  The tendency of grid value (moist static energy and specific humidity) is calculated using below equation,

      

      

 The cloud base level is determined as a lifting condensation level which satisfies,

      

    where,

      

Assume that normalized mass flux is linearly increasing below cloud base,

      

  Using this normalized mass flux, the moist static energy and total water mass at cloud base level can be calculated as,

      

      

  And above the cloud base,

      

      

      

  When , we assume that there is no cloud and the cloud base level is changed to be lower level.

  * Calculation of entrainment rate ()

  The cloud top is the level in where the buoyancy of cloud becomes zero. The buoyancy of cloud() can be represented as,

      

      

      

      

  where,

       : virtual temperature

       : saturation specific humidity

      

       : saturation specific humidity/moist static energy of environment

      

      

  If we assume that

      

 and, we can write again the equation for bouyancy.

      

  Now the entrainment rate can be easily obtained using above equations

      

  where,

      

      

  The entrainment ratio should satisfy below relationship with cloud top height.

      

 Here, precipitation rate() is determined using exponential function ,

        

 then,

      

  Using the entrainment rate (), vertical structure of the normalized mass flux (), moist static energy () and total water mass () in cloud is determined.

  1. normalized mass flux ()

      

      

      

  2. moist static energy ()

      

      

                   

  3. total water mass ()

      

      

                   

                   

  The amount of cloud () is,

      

            

      

  To cloud work function () is defined as a vertical integration of work by bouyancy per unit mass flux ("the kinetic energy generation per unit mass flux", Arakawa and Schbert (1974)), this can be written in equation form,

      

         

  * please see the subroutine 'UPDRF' for the notation.

  As noted in the introduction, the cumulus convection scheme represents the interaction between cloud ensemble and environment (grid scale). Cloud work function means the amount of work which caused by environment and this is consumed by cumulus convection and compensational subsidence in environment. The cloud base mass flux is determined using the condition - the cloud work function is relaxed toward 0 with a specific time scale.

  To consider the relaxation of cloud work function, the tendency of grid value (environment) is calculated using assumed bass mass flux ().

      

      

      

      

   When the calculated cloud work function using is (in the subroutine CLDTST), the cloud base mass flux () is,

      

  where,

      

   The cloud mass flux in one cloud is obtained by multiplying normalized mass flux to cloud base mass flux. And the total cloud mass flux is the sum of mass flux from all clouds.

      

      

  The detraining mass flux is,

      

  Here, it is assumed that detrainment can occur only at cloud top.

  Precipitation rate is also calculated using cloud mass flux.

      

  The cloud water content () is calculated using cloud mass flux () and cloud total water (),

      

   where,

      

    The cloud fraction () is determined using cloud base mass flux () and normalized mass flux (),

      

  where,

      

      

   The effect of the detrainment () at cloud top on the tendency of moist static energy and specific humimdity is calculated as,

      

      

which means,

      

   To maintain the mass balance, the strong updraft at the cloud induces a subsidence in the circumstance. The effect of this adiabatic process on the moist static energy and humidity is,

      

      

  which means,

      

   When the precipitation passes unsaturated air, part of that evaporates. This process is considered in this scheme using saturation specific humidity of environmental air (),

      

  where,

      

      

  Downdraft effect:  Downdraft massflux () is calculated using evaporation rate (),

      

  where,

       : the fraction of downdraft by evaporation to the evaporation rate

  Moist static energy and specific humidity in the air mass which is downdrafting,

      

      

  The tendency of moist static energy and specific humidity by downdraft is,

      

      

  which means,

      

   Considering wet-bulb temperature ,

        

  when the temperature of layer is higher than , the precipitation becomes rain. Or, it added to snowfall.