One could think of brown-ness in literally two dimensions, but a brown channel would be only one dimensional. Ergo, you cannot define a brown channel quite as easily as you may wish.
Even the red channel does not truly define the redness of a color, so you will need to define what you mean by brown channel before anyone could help you. There are many ways one could false color an image to reflect brown-ness of any given pixel. Perhaps you might choose the Euclidean distance in L*a*b* units (often called delta E) from some nominal brown color. Of course, this will give you the same brownness if you are red, or green, or blue (all by the same amount) from that given color. And suppose we move directly out from the neutral axis by k units, as opposed to moving directly into the neutral axis by k units. Surely moving away from neutral would create a color that is MORE brown, but the above definition (in terms of delta E) would say that is not true.
You might think about some path through color space as defining brown. Thus consider the path from white to the point you will define as "brown". (Pick that point carefully. Is it on the gamut boundary? If not, what happens as you move further in that direction?) Now brown-ness is essentially the distance from white along your path. For any color not on that path, project the point orthogonally down onto the path to determine its essential brown-ness. Note that it will help if your definition of the path is a straight line in your color space, as otherwise this orthogonal projection becomes a bit nasty.
Given some time, I might be able to come up with another way to define brown, but this is your question, not mine. Only YOU can define what you mean.
Edit: Given the picture shown in the revised question, I'd suggest a simple scheme. The "brown" spots appear to be composed of dark brown colors, and a lighter, almost yellowish "tan", but not really much in between those two alternatives. So, convert your image into a color space like CIE Lab. Each pixel will now be represented in terms of the corresponding coordinates in Lab space, instead of RGB. Now, choose a nominal brown color, and a nominal yellow, and compute the Euclidean distance in Lab from these two points. If the pixel is within some specified tolerance, then call it brown, or yellow. This is a common solution for your problem.