Mountain warfare is a classical topic and quite interesting. What you really want to do in mountain warfare is to march through valleys, passes and tunnels - or to prevent the same. You need to secure the mountain peaks to do so. This was already reported by Xenophon based on his own experiences more than 2,300 years ago.
And the Greeks were well pleased to see the hills, as was natural considering that the enemy's force was cavalry; when, however, in their march out of the plain they had mounted to the top of the first hill, and were descending it, so as to ascend the next, at this moment the barbarians came upon them and down from the hilltop discharged their missiles and sling-stones and arrows, fighting under the lash. They not only inflicted many wounds, but they got the better of the Greek light troops and shut them up within the lines of the hoplites, so that these troops, being mingled with the non-combatants, were entirely useless throughout that day, slingers and bowmen alike. And when the Greeks, hard-pressed as they were, undertook to pursue the attacking force, they reached the hilltop but slowly, being heavy troops, while the enemy sprang quickly out of reach; and every time they returned from a pursuit to join the main army, they suffered again in the same way. On the second hill the same experiences were repeated, and hence after ascending the third hill they decided not to stir the troops from its crest until they had led up a force of peltasts from the right flank of the square to a position on the mountain. As soon as this force had got above the hostile troops that were hanging upon the Greek rear, the latter desisted from attacking the Greek army in its descent, for fear that they might be cut off and find themselves enclosed on both sides by their foes. In this way the Greeks continued their march for the remainder of the day, the one division by the road leading over the hills while the other followed a parallel course along the mountain slope, and so arrived at the villages.
The higher positions are dominant; that was true with stone slingers and bowmen just as it is today with hand grenades and machine guns. The classic solution is to first employ light infantry to take the mountain peaks, then to proceed in the valley. This was neglected early in the Korean War when South Korean and U.S. forces lacked the infantry to control the mountains. The results were devastating as usual in such cases.
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Mountains and some forms of hilly terrain present an interesting tactical problem. Most vehicles are of little use and there's often little that obstructs the view. The latter is especially true above the tree limit.
Russian unopposed advance through the northern part of South Ossetia (maybe still Northern Ossetia) in 2008. The actual breakthrough happened in a different terrain type.
Open terrain with wide and long fields of fire is usually tank terrain; attacking infantry has marginal survivability on such a terrain if it faces a strong opponent. Mountain areas often have very long unobstructed fields of fire - Tanks and other armoured fighting vehicles are nevertheless much reduced in their value in mountainous areas due to mobility restrictions.
Mountain infantry is therefore in a kind of "1915" situation. It faces extremely powerful hostile fire-power on open terrain and it lacks armour.
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There's not much of a problem with defence; it's almost a dream land for forward observers, mortars and howitzer artillery (the latter being mostly in valleys), at least for the first few kilometres. The calculation of firing solutions was laborious until the 80's, but that problem is mostly gone today.
The terrain makes most methods for counter-artillery targeting ineffective. Sound ranging is confused by echoes, flash sighting is impossible because the barrels are behind mountains and counter-artillery/mortar radars have difficulties with their usual mode of operation as well.
The restrictive terrain makes it also quite easy to guess good spots for harassing fires or marshalling areas.
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The offence problem is this one: How do you advance against a strong opponent on mountainous terrain with little or no concealment & cover on possible routes of approach?
The terrain does often offer enough concealment and even cover for resting and observing troops. The issue is different for advancing troops. You need to move from one rock cover to the next, exposing yourself to observation and fire.
Infantry can negotiate the terrain only slowly and needs to move over long stretches of terrain without enough cover and concealment. The use of smoke for obscuration of movements is also difficult, in part because winds are often strong on mountains.
Combat in Afghanistan confirmed many problems despite the poor marksmanship and ammunition supply of the enemy - who has on top of that almost no indirect high-angle fire support. Combat experiences also confirmed how easily infantry can be pinned down in mountainous terrain.
Classic surprise tactics - offensive movements at night or during poor weather - are of greatly diminished value in the age of infra-red and remote sensors. Such well-timed assaults are still feasible against paramilitary opponents, but they're not really promising against modern military opponents. It's dangerous to negotiate mountain terrain at poor visibility and weather conditions in itself anyway.
Some people discuss the problem of mountain warfare as a hardware choice. Longer-ranged weapons can in theory win a fire-fight at long distances. It's too bad that such longer-range weapons are heavier than normal and require more (and heavier) ammunition than short-range weapons would do. Weight is a painful problem in mountain warfare even with lightweight weapons. The defender can ceteris paribus match and exceed the attacker's fire-power because he doesn't need to move as much.
The long-range weapons path offers no solution for the attacker's troubles, not even for the direct fire fight alone.
Another approach is to forget about tracks, tires and boot movement and hop into helicopters. This is done quite much in Afghanistan.
Hot & high conditions aren't nice to helicopters, strong opponents would have appropriate air defences and conventional mountain warfare would already strain the helicopter force to its limits with logistical tasks. The choice of landing zones is rather limited and powerful defenders could easily cover these LZ with fire. Helicopters are an expensive niche solution, a barely feasible answer to mountain warfare tactical mobility in small wars.
Another approach is the one from the official field manuals. The ones I know basically advise to choose the least terrible route of approach. Use cover & concealment whenever possible, accept to make a detour.
That doesn't solve the basic problem, though. Such attacker-friendly areas would most likely have the due attention of the enemy just like the LZs. The FM advice works best in scenarios with a rather low force density.
Field manuals also emphasize the importance of dispersion, but that's not very convincing either. Dispersion did not keep the infantry capable of offensive actions over flat, open terrain. Why should it suffice on open and much less easily negotiated terrain? Its contribution to a problem solution could only be small.
It's also an option to reject the idea that vehicles cannot handle the terrain and employ them nevertheless. The German allocation of Wiesel weapons platforms with 20mm autocannon or TOW missile launcher to mountain infantry battalions is an example for this approach. The Wiesel is tiny and able to negotiate some bridges and paths up in the mountains. Yet, its ability to go where you want it to go is not reliable. Maybe sometime in the future walking vehicles can negotiate all terrains that can be negotiated with mere boots, but such Sci-Fi won't help us in this decade.
Well, how could a solution for offence in modern mountain warfare look like?
Much of the defender's firepower would consist of support fires. Air power and counter-artillery fires approaches to this challenge would most likely remain very incomplete, but maybe there's another way to take mortars and howitzers out of the equation: It might be possible to cut the communication connections.
Cable communications are possible in mountainous terrain and they're relatively difficult to cut unless you know their position or spend much ammunition, but radio comm is more important anyway.
It might be possible to jam radio comm in mountain warfare.
That's almost guaranteed to fail if only classic radio jamming techniques were used. Hills and mountains are the definition of a shield against jamming attempts. The jammers would usually need to have a line of sight to both emitter and receiver (observer and firing position). Artillery-delivered radio jammers (yes, this stuff exists, and it did so for decades) may be part of the answer (if they can handle the hard mountain surfaces). Aerial jammers may be another part of the answer (preferably cheap medium sized UAVs).
The degradation of opposing (observed) support fires may be part of a solution.
The destruction of forward slope defensive positions by support fires (battalion and higher support) coupled with a suppression of ridge-lines and mountain peaks (by HE) and battlefield interdiction by further fires on bottlenecks behind the opponent's forwardmost mountain peak (fires on road bottlenecks) may help a lot. The idea is basically to minimize the line-of-sight (LOS) combat as well. The less LOS combat, the less casualties among the advancing infantry.
Very light equipment, good physical fitness and adaption to high altitude is of great importance for infantry on mountains. Body armour needs to be limited to partial protection against fragments (light helmet & light "flak vest" at most).
Their mobility on the difficult terrain defines the duration of an attack. The slower they are, the more they're exposed and the longer the need for support fires.
This forms a reinforcing loop: Less duration of support fires allows for more intense support fires and that again should relieve the infantry from some pressure (casualties) - which again allows quicker movement. The limit for this is the speed of the infantry in negotiating the difficult terrain. This speed is being defined by terrain, morale, physical fitness and equipment. Additional equipment can be brought forward once the ridge-line is under control.
Movement speed is of course a huge problem in mountainous terrain, where "everything is slower".
One hour is added for each 300 meters of ascent or 600 meters of descent to the time required for marching a map distance.
FM 3-97.6 Chapter 4-22 (U.S.Army)
Even medium moutaineous terrain (about 1,000 to 3,000 m peak altitude) terribly slows down all movements. Troops with normal flat country training and normal equipment would likely not meet offensive requirements, but maybe properly trained (including stair climbing training instead of normal running) and equipped troops would.
In the end, conquering a mountain peak or ridge as part of an offensive requires either a high tolerance for casualties or a combination of quick infantry with a quite liberal application of combat support.
The new position would allow for good enough observation to marginalize the opponent's activities on the reverse slope, in the next valley and on the counter-slope. It could be necessary to defend it immediately against a quick counter-attack, though.
Such a mountain-by-mountain advance would be terribly "expensive" against a strong opponent no matter how it's done. This is where the airborne idea comes up again. An airborne assault on "rear" mountains could prevent an orderly withdrawal of the enemy to the next ridge-line and it would likely face a less "hot" landing zone than during an assault on the front mountain. (The airborne threat can at the very least motivate the opposing force to develop a deep defence, thus weakening their forward-most defences.)
Maybe there's also an opportunity for a AFV-led assault through the valley at the moment of the opponent's fall-back.
AFVs are extremely vulnerable in mountainous regions not only because they are often limited in their movement; they also expose their weakly protected top to enemies at higher altitudes and main battle tanks lack the gun elevation to fight back effectively. Modern combat vehicles could address the latter two problems with active protection systems that add top protection and with auto cannons (which tend to have a better maximum elevation angle than MBT cannons). Such improvements might suffice to make an armoured thrust through a valley feasible and promising in a situation of turmoil.
The armour attack would serve only one purpose; push forward the culminating point of the attack.
The exploitation of a short transition phase of much-reduced defensive ability is important for the creation of offensive momentum. Some relevant mountainous areas aren't very wide. A single offensive could succeed in breaking through or at least reaching the highest line of mountains from which the later mountain peaks can be observed more easily.
There are of course huge problems with all of this. Friction, morale, isolated pockets of stubborn resistance, areas of enemy-held infantry terrain and unobserved defensive artillery fires, for example.
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The previous thoughts on a possible solution to the mountain offensive problem were about conventional warfare - when an army faces a powerful enemy.
Today's small wars looks very different. Air power can intervene freely, the paramilitary force has almost no mortars and no artillery, the ratio of helicopters to ground troops is quite favourable and the paramilitary force isn't able to establish a deep, known defensive zone without gaps.
This opens many, many more options for the military force. It's kind of discouraging that mountain (and hill) warfare is nevertheless seen as a difficult challenge. Infantry still gets fixed by small arms fire and (counter-) offensive action is often limited to close air support.
That may be in part because of a high risk aversion and because powerful air support is available, but it's certainly no good preparation for wars of necessity; defensive wars against real powers.
(We're lucky that we have a real chance of not seeing such a war for decades, of course. But then again I fear that there's too much stupidity in mankind, and it could get us into real trouble quite soon.)
The snippets about the tactics employed in AFG that I've learned about suggest that the situation could be improved by splitting the military force up into enough separate manoeuvre elements. One or two might still become fixed and suppressed, but others could advance and exploit that the enemy isn't strong enough to maintain control of the terrain at his flanks and rear.
The objective in every fire-fight there should be the encirclement or an ambush on withdrawing enemies; the troops in contact should ensure that no opponent escapes; almost none of them shall be allowed to join another fire-fight against them or their comrades again.
There are only a few thousand enemy combatants, and the huge war machine that got deployed to AFG should really be able to mobilise enough capabilities to win fire-fights decisively.
Merely extracting a platoon out of a fire-fight without casualties and with some claims of enemy WIA and KIA is not sufficient; that's pretty much a draw and will likely lead to a more refined attack by the enemy on a later date.
March separated - encircle or ambush - destroy trapped enemies with support fires. The "destroy with support fires" part may be substituted with "capture prisoners", of course.
The result could be that the option of infantry-on-infantry combat in mountainous or hilly terrain would be scratched from the enemy's active repertoire.
Instead, warfare on mountains and hills is still a huge problem, most opponents escape from fire-fights. Seriously; modern, trained and lavishly equipped forces that struggle to achieve decisive tactical victories against a rag-tag militia despite a ~10:1 manpower advantage and a ludicrous budget and material advantage are an embarrassment. They would likely fail catastrophically in a conventional war mountain offensive against a strong opponent.
I don't blame the small units in contact; it's the whole concept of fort garrisons doing little more than a few raids and patrols that fails embarrassingly in AFG. Isolated small infantry patrols can easily be pinned down and limited to calls for fire support.
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Mountain warfare may today be at a point where offence is much, much stronger than defence. It may be true that there's no way how an offensive could proceed through mountainous terrain against a strong defence without terrible costs. The prospect for rapid advances without major assistance by the opponent's failures is marginal.
Mountain warfare against paramilitary forces should be relatively easy given the extreme asymmetry of support and training, though.