Brine-wetted snow

Brine-wetted snow is modelled as a dry snow background with inclusions of brine.

Consequently, scattering from snow grains is ignored. This is a crude approxi-mation, but is assumed to be adequate for the purpose of this thesis considering the relatively high permittivity of brine compared to air and pure ice. The per-mittivity of dry snow is calculated from the empirical relation in equation 5.16.

equation 3.3 is used for the permittivity of brine. The volume fraction of brine is calculated as (Drinkwater & Crocker 1988, equation 4):

Vi= ρbsVb

(1−Vb)ρpi+Vbρb

(5.17) whereρbs,ρpi, ρb are the densities of the brine-wetted snow, pure ice and brine respectively. The latter two are given by equation 3.1 and equation 3.3, respec-tively. Vb is the volume fraction of brine relative to the pure ice volume. This is calculated assuming that the phase relation between pure ice and brine behaves as in standard sea ice and thus can be approximated by equation 3.6. Again, the temperature is derived from the above and below layers assuming the thermal conductivity being the same as for dry snow (equation 5.15).

An example of the resulting effective permittivity as a function of density and temperature according to SFT is shown in figure 5.4. The calculations are compared to the empirical equations for dry snow by Hallikainen et al. (1986) and Tiuri et al. (1984). Overall, both the real and imaginary parts are higher in the brine-wetted snow as compared to the empirical equations for dry snow.

Both the real and imaginary part deviate most from the dry snow case at high temperatures, due to relatively high volume fractions of brine.

As for dry snow and sea ice, the correlation of the top surface is assumed close to exponential with a GPLp-value of 2.1. In summary, the following independent parameters are:

Dry vs. brine-wetted snow

Figure 5.4: The relative permittivity of dry and brine-wetted snow (ε=ε^′−iε^′′) plotted as a function of snow density for the top two plots and snow temperature for the bottom two plots. The dry snow (solid blue line) is computed with the empirical relation in equation 5.16 and for the brine-wetted snow (dashed red line) the SFT is used with brine inclusions in a dry snow background. The frequency is 5.4 GHz (C-band), both the minor and major correlation lengths are 0.5 mm, the salinity of the brine-wetted snow is 40 ppt. For the top plots the temperature is -7^◦C and for the bottom ones the density is 400 kg/m³.

ρbs = brine-wetted snow density Sbs = brine-wetted snow salinity lbs = Brine droplet correlation length σz ,bs = Surface RMS-height

σs,bs = Surface RMS-slope dbs = Snow thickness

5.3.3 Sea ice

Sea ice is modelled as a two phase medium consisting of a pure ice background with inclusions of liquid brine. As such, scattering from air filled structures such as bubbles or empty drainage channels is ignored. This is a reasonable approximation for sea ice with a relatively high salinity, due to the considerably larger permittivity difference between brine and pure ice compared to air and pure ice (see section 3.1). For low saline ice, such as multi-year ice (MYI) or ice formed in brackish water such as the Baltic, inclusions of air are significant. In this thesis, such ice types are however not considered.

The single Debye relaxation relation stated in equation 3.3 is used for the permittivity of brine and the empirical relation stated in equation 3.2 is used for the permittivity of pure ice. The volume fraction of brine is calculated assuming standard sea ice (see section 3.2) using the relation in equation 3.6. In summary:

εb = Pure ice background permittivity from equation 3.2 εi = Brine inclusion permittivity from equation 3.3 Vi = Brine volume fraction from equation 3.6

Using this parametrisation,εb, εi and Vi are determined by the temperature T and salinityS. The temperature is calculated from the layers located above and beneath the current layer, using the thermal conductivity (Thomas & Dieckmann 2009, equation 2.14, page 48):

κsi=κpi+ 0.13Ssi

T² (5.18)

where Ssi is the salinity of the sea ice and T is the temperature in ^◦C and the pure ice thermal conductivity is given by (Thomas & Dieckmann 2009, equation 2.11, page 48):

κpi= 1.16×(1.91−8.66×10⁻³T+ 2.97×10⁻⁵T²) (5.19) The effective permittivity is calculated using SFT. An example is shown in figure 5.5 where the SFT results are compared to the empirical relation by Vant

et al. (1978):

ε^′_si=a0+a1Vb (5.20a) ε^′′_si=b0+b1Vb (5.20b) where a0,1 and b0,1 are empirically determined coefficients (see B.4) and Vb is the volume fraction of brine. The coefficients are given for first-year ice (FYI) and MYI, separately. With equations 3.1 and 3.6 forVb and the pure ice density ρpi, the empirical sea ice permittivity becomes a function of the sea ice salinity Ssi and temperature T. Note that the empirical relation does not account for anisotropy in the sea ice.

Overall, the SFT results are similar to the empirical relation. The real part of the permittivityε^′_siis generally higher according to SFT, while the imaginary partε^′′_siis lower. The deviation in the real part is higher at high salinities, while the imaginary part deviate more at lower salinities. This may be a result of neglecting air inclusions which have a low permittivity (thus lowers the real part) but scatter (thus increases the imaginary part).

Regarding the roughness of the ice surface, systematic measurements on scales relevant for microwaves are scarce. Johansson (1988) and Drinkwater (1989) how-ever find that the assumption of a Gaussian height distribution is accurate. Kim et al. (1985), Johansson (1988) and Dierking (1999) indicate that the exponential correlation function is realistic for undeformed sea ice. Here, the GPL correlation function is therefore used withp= 2.1. The surface is then close to the exponen-tial (p= 2) but has a well defined RMS-slope (see section 4.1 for details).

In summary, this results in the following independent model parameters for the sea ice layer:

Ssi = Bulk salinity

lρ,si = Brine minor correlation length Esi = Brine elongation (=lz,si/lρ,si) ψsi = Brine inclusion tilt angle σz ,si = Surface RMS-height σs,si = Surface RMS-slope dsi = Ice thickness