I'm specifically trying to work out how far away an observer could see the dust cloud of 5000 horsemen (say, a Mongol or Turkic army) over a desert. I know it's 35x(root of the observer's height + root of the dust cloud's height) km for horizon calculations. But I don't know what kind of height the dust cloud would be. But perhaps there are some histories of ancient battles that talk about how far away the dust cloud of an approaching army was? It doesn't have to be horsemen specifically, anything would be helpful. Google hasn't given me anything helpful, so I'm turning to military historians for help.
This source, originally published in the Autumn 2004 edition of MHQ: The Quarterly Journal of Military History, doesn't give specific distances but highlights several instances of dust clouds and their impact on ancient battles. http://www.historynet.com/military-technology-using-a-cloud-of-dust-in-ancient-warfare.htm
[Livy]recorded how in 306 b.c., Samnites were besieging a Roman army led by Counsel Publius Cornelius Arvina when they learned of the approach of a second enemy force, led by Counsel Quintus Marcius Tremulus. Briefly turning their backs on Cornelius, they decided to ambush the approaching troops:
Thirty thousand Samnites were killed, according to Livy, in a defeat that was to a large degree attributable to clouds of dust betraying their plans.
In addition to indicating the presence of the enemy, dust was used, albeit imperfectly, to calculate the size, nature, and movement of opposing forces. There has been some speculation that high, thin clouds of dust correlated with the presence of cavalry, while dense, lower clouds could be associated with infantry.
Of course, the tendency to interpret clouds of dust in predictable ways could be used to advantage by an enterprising commander. Livy recounted a notable incident from Rome’s 293 b.c. campaign against the Samnites. The Romans had divided their forces, with Papirius outside Aquilonia and Carvilius twenty miles away outside Cominium, and planned to launch coordinated attacks against the Samnite towns. Before engaging the enemy forces at Aquilonia, Papirius had a legate and three auxiliary cohorts lead his army’s mules along a hidden route to a nearby hill. During the subsequent fierce battle, on the Samnites’ flank appeared a cloud of dust as though an enormous column of men were in motion to attack, Livy wrote. It was, however, the auxiliaries, who were riding the mules while dragging leafy boughs behind the beasts. Confusion swept through the Samnite ranks, as well as through the Romans’, as it appeared that a massive army had raised the cloud of dust. Papirius did his best to reinforce that impression by shouting loudly, Cominium has fallen, my victorious colleague is coming on the field, do your best to win the victory before the glory of doing so falls to the other army! He then ordered his troops to open ranks, and the Roman cavalry swept through and routed the enemy.
While dust clouds often betrayed enemy movements, they could also be important in finding a battle already underway. Thucydides, in his fifth century b.c. account of the Peloponnesian War, observed how the Corinthians stationed at Cenchreae correctly interpreted the dust clouds they saw as indicating the place of battle. As a result they hurried to the rescue of their comrades there.
Similarly, Caesar, during his first British expedition, was alerted to his troops’ peril by a cloud of dust. After he had sent a legion out from his camp to forage for corn, guards reported to him the presence of a large cloud of dust in the direction that the legion had headed. According to the great commander’s own account: Caesar guessed the truth — that the natives had hatched a new scheme. He set out with troops toward the dust cloud and found the legion hard-pressed by Britons, who broke off their attacks upon his arrival.