To delete the alert history that appears on the timeline in the Ambient Weather Network, follow these steps:

1. Access Your Account: Go to the Ambient Weather Network website and log into your account.

2. Navigate to the Timeline: Once logged in, locate the timeline section on your dashboard, which displays the history of alerts.

3. Current Limitations: As of now, Ambient Weather has indicated that the alert history on the timeline cannot be deleted. This means that while you can manage or delete alerts you have set up through the alerts panel (found under /alerts), you won't find an option to delete historical alerts from the timeline itself - learn more.

4. Stay Updated: For any changes regarding this feature or future updates, keep an eye on the official Ambient Weather forums or updates from their support team.

As it stands, if the alert history cannot be cleared, you may want to provide feedback to Ambient Weather requesting this functionality, as user input can influence future updates to their interface.