Migrating Search Application from SharePoint 2010 to SharePoint 2013

-- Take the backup of the Search Admin Database from SharePoint 2010.
--   Copy the backup to the SharePoint 2013 database server.
-- Restore the database.
-- Once the restoration of the database is done, refresh the root node to see the database.
-- Go to the SharePoint 2013 server and open the PowerShell console as administrator and run the below command.

New-SPServiceApplicationPool -Name SPSearchServiceAppPool –Account “<SP 2013 Service Account>”

-- Once the SPSearchServiceAppPool is created

-- Run the below script 

        $appPool = Get-SPServiceApplicationPool -Identity "SPSearchServiceAppPool"

        $searchInst = Get-SPEnterpriseSearchServiceInstance -local

        Restore-SPEnterpriseSearchServiceApplication -Name "Search Service Application" -applicationpool $appPool -databasename "<Search Admin DB Name>" -databaseserver         "<DB Server Name>" -AdminSearchServiceInstance $searchInst

-- Once the DB is attached to the Application

-- Run the below script

        $ssa = Get-SPEnterpriseSearchServiceApplication -Identity "Search Service Application"

        New-SPEnterpriseSearchServiceApplicationProxy -Name "Search Service Application" -SearchApplication $ssa

-- For Verification of the Migration, you should be able to see a new Search Service Application created in the list of Service Applications in CA.

-- And you should be able to see the SharePoint 2010 DB Name shown under 'Search Application Topology' in CA.