Sorting query results with a sort by select box

I am running query's on my batsmen table and would like to sort the results A-Z and Z-A as well as High to Low on top score by using a select option box without reloading the page. I am having a problem working out how to get the AJAX js working with the php and select box any ideas on how to do this, i can't find any useful tutorials. I have this code so far:

Controller(PHP):

        $batsmenQuery = Batsmen::where('approved', '=', 1);

       switch ($request->SortbyList){
            case 0:
                $batsmenQuery = $batsmenQuery->orderBy('name', 'DESC');
                break;
            case 1:
                $batsmenQuery = $batsmenQuery->orderBy('name', 'ASC');
                break;
            case 2:
                $batsmenQuery = $batsmenQuery->orderBy('hs', 'ASC');
                break;
            case 3:
                $batsmenQuery = $batsmenQuery->orderBy('hs', 'DESC');
                break;
                default:
                    $batsmenQuery = $batsmenQuery->orderBy('name', 'DESC');


        }

        $batsmen= $batsmenQuery->paginate(40);

HTML:

            <div class="row">
                <div class="sort">
                    <select name="SortbyList" id="SortBy">
                        <option value="0">A to Z</option>
                        <option value="1">Z to A</option>
                        <option value="2">Highest Score</option>
                        <option value="3">Lowest Score</option>

                    </select>
                </div>
            </div>

javascript:

     $('#SortBy').on('change', function(e){

     });

Any ideas on how to make this work?

via Chebli Mohamed