I'm attempting to construct a world such that a hot desert like the Sahara exists in the region outlined in yellow. The same map depicting prevailing winds (thick arrows) and ocean currents (thin arrows) I'm working under the following constraints:

1. the desert must be located on the eastern half of the continent
2. the desert must directly border the ocean to the south (it cannot be a solely interior desert)
3. the desert should believably arise through natural means (geography/climate) and not artificially, such as via magic.

I've added a very tall mountain range (think Himalayas) to the east to produce a rain shadow effect, placed the region such that the prevailing winds will blow off shore, and oriented a cold ocean current on the western side.

Are these factors sufficient to produce a desert in this region, and how far could it reasonably extend? If not, what changes can I make to achieve this?

I'm attempting to construct a world such that a hot desert like the Sahara exists in the region outlined in yellow. The same map depicting prevailing winds (thick arrows) and ocean currents (thin arrows) I'm working under the following constraints:

1. the desert must be located on the eastern half of the continent
2. the desert must directly border the ocean to the south (it cannot be a solely interior desert)
3. the desert should believably arise through natural means (geography/climate) and not artificially, such as via magic and over-farming

I've added a very tall mountain range (think Himalayas) to the east to produce a rain shadow effect, placed the region such that the prevailing winds will blow off shore, and oriented a cold ocean current on the western side.

Are these factors sufficient to produce a desert in this region, and how far could it reasonably extend? If not, what changes can I make to achieve this?