It depends upon the risks, factors, and age of the patients. If they have hypercoagulable states then the patient needs anticoagulation otherwise, aortic atheroma is generally "white" clot which is made up of platelets and cholesterol and hence, treatment would be anti-platelets and cholesterol lowe...

Aortic thrombus or complex atheroma >4mm or hypermobile need anticoagulation and hypercoagulable work up.  If no hypercoagulable condition is identified then after 4-8 weeks reimage and, if resolved, de-escalate to antiplatelet.