Update textArea while processing JSF

Question

I have a component textArea inside my XHTML page that should show the 'LOG' of a proccess. See:

<h:inputTextarea rows="20" value="#{myMB.logVar}"
           id="iptTextAreaLog" disabled="true"
           style="color:white;background-color: black;width: 95%;">
</h:inputTextarea>
<h:commandButton value="atualiza log" id="cmbButtonAtualizaLog">
           <f:ajax render="@form" execute="@this"/>
</h:commandButton>

This button is called from a javascript setInterval that triggers a update in textArea.

I have a method that update the logVar variable:

public void iniciarCarga() {
        new Runnable(){    
            @Override
            public void run() {
                service.enviar(getSubject().getLogin(), valor, logVar);
            }
        }.run();    
 }

This is the javascript code that call my commandButton:

setInterval(function(){
     $('#cmbButtonAtualizaLog').click();
},1000)}

The method inicarCarga() updates the logVar variable and this value should be shown inside textArea, but this don't happen.