RHS    1999/06/04

Calculation of Surface Relative Humidity


Constants
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TKF  (K)      = freezing point of water            =     273.16
ELHE (J/kg)   = latent heat of vaporization at 0 C = 2500000
ELHS (J/kg)   = latent heat of sublimation at 0 C  = 2834000
SHCV (J/kg*C) = specific heat capacity of vapor    =    1911
SHCW (J/kg*C) = specific heat capacity of water    =    4185
SHCI (J/kg*C) = specific heat capacity of ice      =    2060
RDRY (J/kg*C) = gas constant for dry air           =     287
RVAP (J/kg*C) = gas constant for water vapor       =     461.5
PPVSF (Pa) = saturated partial pressure of water vapor at 0 C = 610.571


Model variables from CD-ROM
---------------------------
TS (C)          = surface air temperature 
QS (g/kg)       = surface air specific humidity 
PS (mb-1013.25) = surface pressure 


Calculation of surface relative humidity
----------------------------------------
ELHV (J/kg) = latent heat of vaporiztion =
            = ELHE + (SHCV-SHCW)*TS   when   TS > 0
            = ELHS + (SHCV-SHCI)*TS   when   TS < 0

PPVS (Pa) = saturated partial pressure of water vapor as function of TS
          = PPVSF * exp[ELHV*(1/TKF - 1/(TS+TKF))/RVAP]

PPV (Pa)  = partial pressure of water vapor =
          = RVAP*(QS*.001)*(PS*100+101325)/[RDRY+(RVAP-RDRY)*(QS*.001)]

RHS (%) = 100*PPV/PPVS


Model approximations of surface relative humidity
------------------------------------------------
PPVSF (Pa) = 610.71

ELHV (J/kg) = ELHE   when   TS > 0
            = ELHS   when   TS < 0

PPVS (Pa) = PPVSF * exp[ELHV*(1/TKF - 1/(TS+TKF))/RVAP]
PPV  (Pa) = RVAP*(QS*.001)*(PS*100+101325)/RDRY
RHS  (%)  = 100*PPV/PPVS


Systematic error in calculation of relative humidity from monthly means
-----------------------------------------------------------------------
Because the graph of PPVS vs TK is concave, the relative humidity
calculated from monthly mean temperature and specific humidity will be
systematically higher than the monthly mean relative humidity.  For
example, suppose that the temperature at a given grid box is 240 (K)
for half the month and 260 (K) for the other half of the month, that
the relative humidity is always 100%, and that the pressure is 100000
(Pa).  When the temperature is 240 (K), the saturation specific
humidity will be .169116 (g/kg).  When the temperature is 260 (K), the
saturation specific humidity will be 1.216900 (g/kg).  The monthly mean
temperature will be 250 (K) and the monthly mean specific humidity will
be .693008 (g/kg).  The relative humidity calculated from these monthly
mean values will be 146.8%.