Order in Search Page without Relevance when ordering by price

We have a situation where in the search page (not in the ajax search heade) when someone sorts by Price Desc or Price Asc, the results are also being ordered/grouped by the relevance of the search terms as well.

So just to explain myself better. If someone searches for ‘batteries’ and there are 10 results with different prices and 5 of them have ‘batteries’ in the title the other 5 do not have.

After someone sorts the results by price asc first we are getting the 5 results that have ‘batteries’ in the title ordered by price asc after those 5 products we have the other 5 products that do not have ‘batteries’ in the title in price ASC. Therefore the price is not ascending from the first product till the last.

This makes sense for us when doing the ajax search filter in the header however we do not want the relative ordering/sorting on the search page as well.

Is there a way to have the search page only ordered by price without any relevance whilst keeping the same relevance ordering for the ajax search filter?