Different corrections of relative humidity for subfreezing conditions in GC-Net and PROMICE

[BaptisteVandecrux]

Different expressions for the saturation vapour pressure over water and ice.
This step occurs when the relative humidity given by the instrument with regards to water is adjusted for subfreezing conditions, meaning with regards to ice.

In GC-Net:

Lv = 2.5001e6;  % H2O Vaporization Latent Heat (J/kg)
Ls = 2.8337e6;  % H2O Sublimation Latent Heat (J/kg)
Rv = 461.5;     % H2O Vapor Gaz constant (J/kg/K)
if T < 0
    TCoeff = 1/273.15 - 1/(T+273.15);
    Es_Water = 6.112*exp(Lv/Rv*TCoeff);
    Es_Ice = 6.112*exp(Ls/Rv*TCoeff);
    RH = RH * Es_Water/Es_Ice; % where the adjustment is applied
end

In PROMICE (as far as I know):

T_0 = 273.15;
T_100 = 373.15;
% eps is the ratio of the molecular weights of water and dry air
eps = 0.62198; 

% GOFF-GRATCH 1945 equation
es_wtr = 10.^( ...
    -7.90298*(T_100./T - 1) + 5.02808 * log10(T_100./T) ...   % saturation vapour pressure above 0 C (hPa)
    - 1.3816E-7 * (10.^(11.344*(1.-T./T_100))-1.) ...
    + 8.1328E-3*(10.^(-3.49149*(T_100./T-1)) -1.) + log10(1013.246) );

es_ice = 10.^( ...
    -9.09718 * (T_0 ./ T - 1.) - 3.56654 * log10(T_0 ./ T) + ...
    0.876793 * (1. - T ./ T_0) + log10(6.1071)  );   % saturation vapour pressure below 0 C (hPa)
 
freezing = T>=T_0; 
RH(freezing) = RH(freezing) ./ es_ice(freezing) .* es_wtr(freezing); % where the adjustment is applied
¨

References for saturation vapour pressure:

*Goff, J.A. and Gratch, S., 1946. Low pressure properties of acontribution to the classical theory. Quart. J. Roy. Meteor. water from-160 to, 212, pp.95-121.
*Lowe, P.R., 1977. An approximating polynomial for the computation of saturation vapor pressure. Journal of Applied Meteorology, 16(1), pp.100-103.
*Weiss, A., 1977. Algorithms for the calculation of moist air properties on a hand calculator. Transactions of the ASAE, 20(6), pp.1133-1136.
*More formulations and WMO standards: https://sciencepolicy.colorado.edu/~voemel/vp.html
*
cs500 documentation

Script for the plot above:

T = -30:0;

%% PROMICE way
T = T+273.15;
T_0 = 273.15;
T_100 = 373.15;
% eps is the ratio of the molecular weights of water and dry air
eps = 0.62198; 

% GOFF-GRATCH 1945 equation
es_wtr = 10.^( ...
    -7.90298*(T_100./T - 1) + 5.02808 * log10(T_100./T) ...   % saturation vapour pressure above 0 C (hPa)
    - 1.3816E-7 * (10.^(11.344*(1.-T./T_100))-1.) ...
    + 8.1328E-3*(10.^(-3.49149*(T_100./T-1)) -1.) + log10(1013.246) );
es_ice = 10.^( ...
    -9.09718 * (T_0 ./ T - 1.) - 3.56654 * log10(T_0 ./ T) + ...
    0.876793 * (1. - T ./ T_0) + log10(6.1071)  );   % saturation vapour pressure below 0 C (hPa)

%% GC-Net way
T = T-273.15;

Lv = 2.5001e6;  % H2O Vaporization Latent Heat (J/kg)
Ls = 2.8337e6;  % H2O Sublimation Latent Heat (J/kg)
Rv = 461.5;     % H2O Vapor Gaz constant (J/kg/K)

TCoeff = 1./273.15 - 1./(T+273.15);
Es_Water = 6.112*exp(Lv./Rv.*TCoeff);
Es_Ice = 6.112*exp(Ls./Rv.*TCoeff);

%% plotting
set(0,'defaultfigurepaperunits','centimeters');
set(0,'DefaultAxesFontSize',15)
set(0,'defaultfigurecolor','w');
set(0,'defaultfigureinverthardcopy','off');
set(0,'defaultfigurepaperorientation','portrait');
set(0,'defaultfigurepapersize',[29.7 16]);
set(0,'defaultfigurepaperpositionmode','auto');
set(0,'DefaultTextInterpreter','none');
set(0, 'DefaultFigureUnits', 'centimeters');
set(0, 'DefaultFigurePosition', [.25 .25 [29.7 16]-0.5]);
set(0,'defaultfigurepapersize',[29.7 16]);

f = figure;
hold on
plot(T, es_wtr./es_ice,'LineWidth',2)
plot(T, Es_Water./Es_Ice,'LineWidth',2)
axis tight square; box on; grid on;
xlabel('Air temperature (^oC)','Interpreter','tex')
ylabel({'Correction factor applied to RH','in subfreezing conditions (-)'})
legend('PROMICE','GC-Net')
print(f,'RH_cor_GCNet_PROMICE','-djpeg')

[BaptisteVandecrux]

Other formulation used by Jason:

; --------------------------------- Saturation Vapor Pressure terms
; Es is the saturation vapor pressure, calculated using the formulation of
; Bolton (1980), accurate to within 0.1% over the +/- 30∞C range
; Bolton, D. 1980. The computation of equivalent potential temperature,
; Mon. Wea. Rev., 108, 1046-1053. Eq. 10.

; ------ Saturation Vapor Pressure
  e_t0=6.112
  Es=e_T0*(exp((17.67*T)/(T+243.5)))
; ------ Vapor Pressure
  e=(Es*(rh/100.))