Upgrade from Drush6 to Drush 7 on OSX running Acquia Dev Desktop

Question

My question is really as it is stated in the title.

I use MacOSX and the Acquia Developer Desktop for running my Drupal sites that I develop locally.

I'd love to try out the new Drush 7, but I can't get it to 'upgrade' from Drush 6 to Drush 7.

I've tried:

1. cd into my /Applications/dev desktop
2. removing the drush folder
3. replacing it with a fresh .zip from GitHub

Ofcourse, when entering "drush" I get the message that I should have used Composer instead.

Installing with composer seems to work, but it doesn't install Drush 7 in my Application/Dev Desktop(/drush) folder, so it's of no use to me.

How do I force Composer to put drush in this specific directory? I get confused by all the configuration files and what to put in them. Is there something like > composer install drush --right-here ?

EDIT

After Clide's helpful answer, I get Drush version 7 running. I checked the version using "drush version". However, when I try a real drush command ( drush en -gmap, in this case), I get the notorious "Command pm-enable needs a higher bootstrap level to run" error.

I invoke the command line from within the Acquia application, so the PATH should be correct. I also tried specifying the URI, and also that looks okay. I'll copy some of the output, if that helps.

 Drupal version         :  7.26                                                         
 Site URI               :  http://(sitename).local:8083                            
 PHP executable         :  /usr/local/bin/php                                           
 PHP configuration      :  /usr/local/etc/php/5.3/php.ini                               
 PHP OS                 :  Darwin                                                       
 Drush version          :  7.0-dev                                                      
 Drush temp directory   :  /tmp                                                         
 Drush configuration    :                                                               
 Drush alias files      :                                                               
 Drupal root            :  /Users/(myname)/Documents/(dev folder)/(sitename)
 Site path              :  sites/(sitename).local

Answer 1

In this context Composer is used to fetch the libraries Drush needs, it doesn't download Drush itself. Running the install process in the Drush root should do the trick:

git clone https://github.com/drush-ops/drush.git drush 
cd drush
composer install

and if you don't already have it

ln -s /path/to/drush/drush /usr/local/bin/drush # or wherever