Figure 1.
Schematic of the pressure term P(h; b) in (3): the modified pressure above the threshold is lower than the standard pressure , thus forcing the fluid to rise where .
Figure 2.
Time evolution of the height profile for the standard shallow water case I (left), case II with convection and no rain with (middle) and case III with convection and rain for finite (right). Non-dimensional simulation details: .
Figure 3.
Hovmöller plots for the Rossby adjustment process with initial transverse jet: case I (left), II (middle) and III (right). From top to bottom: h(x, t), u(x, t), v(x, t) and r(x, t). Non-dimensional simulation details: same as Fig. 2.
Figure 4.
Evolution of h and r for the Rossby adjustment process with initial transverse jet: case I (left), II (middle) and III (right). Top row: Hovmöller plots for h. Subsequent rows: profiles of h (black line; left axis) and r (blue line; right axis) at different times denoted by the dashed lines in the top row. Non-dimensional simulation details: same as Fig. 2.
Figure 5.
Hovmöller plots for the Rossby adjustment process with initial transverse jet, highlighting the conditions for the production of rain: case III. From left to right: , , and r(x, t). Non-dimensional simulation details: same as Fig. 2.
Figure 6.
Top row: Hovmöller diagram plotting the evolution of the departure from geostrophic balance : light (deep) shading denotes regions close to (far from) geostrophic balance. Subsequent rows: profiles of fv (red) and (black) at different times denoted by the dashed lines in the top figure. For case I (left), II (middle), and III (right). Non-dimensional simulation details: same as Fig. 2.
Figure 7.
Flow over topography (, and ): profiles of , b (black; left y-axis), exact steady-state solution for the SWEs (red dashed; as derived in Appendix 4) and rain r (blue; right y-axis) at different times: case I (left), II (middle) and III (right). The dotted lines denote the threshold heights . Non-dimensional simulation details: .
Figure 8.
Hovmöller plots for flow over topography (), highlighting the conditions for the production and subsequent evolution of rain: case III. From left to right: , and r. Non-dimensional simulation details: same as Fig. 7.
Figure 9.
Same as Fig. 7 but with two orographic ridges: , , and . Non-dimensional simulation details: same as Fig. 7.
Figure 10.
Same as Fig. 8 but with two orographic ridges. Non-dimensional simulation details: same as Fig. 7.