Example: I scythe a 6-in wide strip through a cover of white clover and other volunteers. I sow carrot seeds, water, and place a sheet of cardboard over the top. 7 days later (carrots usually germinate within 7-10 days) I remove the cardboard and let the carrots do their thing.
The roots of the living mulch are still present and alive. Clover in particular forms a dense mat of shallow roots. Will this mechanically inhibit carrot growth the way dense soil would?
I've never tried it. If you give it a try, I'd be very interested in hearing about your results.
In that I lightly till the surface , tilling in the partially decomposed mulch plus added amendments between crops, I've never planted directly into a living cover crop. This year I've been experimenting using sweet potatoes as a living mulch planted atop a layer of grass clippings. So far it seems to be working with the taro, sugar cane, and bananas. I'm still seeing some grasses coming through. But with this year being unusually wet, I can't say if it has a positive effect on retaining soil moisture. So I'll have to keep experimenting.
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Since a carrot is grown specifically for its root, I would think that you would want to remove all root competition around the carrots. Most annuals have shallow root systems in comparison to most perennials, so, once again the competition would most likely be harmful. Annual plants have a very short life span, where they must accomplish everything from germination to reproduction. That is why people tend to 'baby' annuals much more so than perennials. They need all of the help that they can get. Anything that hinders any one phase of their life cycle is bound to have a harmful effect on production.
Root penetration is probably more governed by the soil type. The above-ground competition with other annuals or perennials would be more a problem of competition for light I think.
I wouldn't put carrots and clover together, unless I intended to just harvest the carrot greens, which would be sort of tedious work.
I put carrots with lambsquarters this year and we harvested the lambsquarters when they were a foot high and the carrots were beginning to take over. That worked pretty well and I think the position and light was more of an issue with the root size (which was generally good but not great). These were both grown in compost since my soil doesn't like carrots.
So maybe it depends on the 'other crop' you're growing next to the carrots.