Here, give this a try:
select name, city, serial_number from
(select name, city
, (case when @city <> city then @citynum := 1 else @citynum := @citynum + 1 end) as serial_number
, (case when @city <> city then @city := city end) as v
from tbl_cities, (select @citynum := 0, @city := '') as c
order by city) as cities
Here is a link to your example running: http://sqlfiddle.com/#!2/615f5/1.
You will need to change 'tbl_cities' to the name of your table.
I could not get it to save as a view or use in an update statement as it uses variables. One quick way would be to use the select statement above to create a new table and run periodically as new data is added.
drop table if exists tbl_cities_serial;
create table tbl_cities_serial(select name, city, serial_number from
(select name, city
, (case when @city <> city then @citynum := 1 else @citynum := @citynum + 1 end) as serial_number
, (case when @city <> city then @city := city else @city end) as v
from tbl_cities, (select @citynum := 0, @city := '') as c
order by city) as cities)