Baroclinic waves and frontogenesis with an embedded zone of small moist symmetric stability

Three-dimensional semigeostrophic moist baroclinic waves and frontogenesis are studied both analytically and numerically. It is found that a slantwise zone of small (either unconditional or quasi-conditional) moist symmetric stability embedded in a dry and baroclinic basic state traps the energy of baroclinic waves, forming a strong meridional structure with maximum wave amplitude in the moist zone and exponential decay of the wave amplitude away from the moist zone. The unstable waves absorb energy from both the moist and dry baroclinic basic state, but the strongest energy conversion is in the moist zone, so growth rates decrease with the width of the moist zone. As the moist zone becomes infinitely narrow (wide) the model degenerates into a dry (moist) Eady model. The smaller the moist symmetric stability, the larger the growth rate and the wavenumber of the most unstable wave. The valid range of the model parameters depends on the semigeostrophic approximation, which sets a limit to the smallness of the moist symmetric stability. In semigeostrophic space, the moist zone changes shape due to geostrophic deformation. As the wave develops in real space, the surface pressure low (high) shifts northward (southward) due to geostrophic deformation, tightens (expands) due to ageostrophic convergence (divergence), and goes on to form a coupled warm-cold frontal system. A numerical model of the semi-Lagrangian finite element method is developed. The method is shown to be accurate in computing wave growths and efficient in resolving frontal structures, although the semigeostrophic approximation is found to become poor or even invalid locally in the narrow (meso-β-scale) cold frontal region.