If the trees were originally planted in a square grid, disking can be done in both directions. Cover Crops protect the soil and prevent invasion by weeds. Timothy is inexpensive to plant, stays reasonably short and can be mowed once a year, after it forms a seed head. Perennial ryegrass stays reasonably short and can be mowed often. However, it is expensive to plant, is subject to winter kill, and low, frequent mowing often allows invasion by dandelions.
Orchard grass is inexpensive to plant, stays relatively short and is moderately shade tolerant.
Tall fescue and red fescue are often used as cover crops but may be allelopathic to black walnut roots. Mowing is a well-known weed control method and is effective between rows. However, because it does not kill the weeds, the grower must monitor weed growth regularly and mow often. Mulches may be applied with varying success. Plastic mulches are difficult to keep in place and are not very long lasting.