I made a dummy example here:
require(dplyr)
expand.grid(x = 1:20, y = 1:2) %>%
mutate(z = rnorm(length(x))) %>%
ggplot()+geom_tile(aes(x=x, y=y, fill=z))+
scale_y_continuous(expand=c(0,0),breaks=NULL,limits=c(0,3.6))+
scale_fill_continuous(low="darkgreen", high="white")+
coord_polar(start=-1*pi/2, direction=1)+
theme_bw()+
theme(panel.grid.major = element_blank(),panel.grid.minor = element_blank())

You are on the right track with arguments limitsand expand scale_yyou just need to find out where the actual lower bound is. To do this, let me build the same set without coord_polarand without yours scale_y.

, y=0.5. , y, height geom_tile ( 1). y, .
