the idea came from Sri's kaju mountain trip story and just some scene in mountain poped up in my mind with a heart to heart conversation of them on why they landed up in this situation
its completely my take sorry if i hurt anyone but take this as my pov only you can dislike this as your right but i can't stop myself from expressing this
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So because of fighting in royal court of Hastinapur both Karna and Arjun got punishment of going on a trip to Himalaya together alone. Even king of Indraprastha Yudhisthir too supported Dhritarastra's this decision to maintain peace and Bheeshma added one more condition on the trip that after the trip they have to comeback together unharmed and if one is back then he will be found guilty of the murder of other and will be beheaded also they have to come back unharmed without losing any part of body causing by other's weapon then too they will be punished. Also vidura added that they are not allowed to use any celestial weapon in Himalaya as himalaya is worshipped as God of mountains and is father of Devi Parvati so Parvati will get angry and curse if they use celestial weapon and harms himalaya's divine nature.
so with all such conditions they started the journey with worried face of both Pandava and Kaurav, Arjun missed krishna even more in such situation its only Subhadra who assured him that he will surely find some reason to enjoy in the trip.
Anyway regarding their nature you can assume how the verbal fight could take place in initial stage. Both tried to take two different ways but soon realized being alone in Himalaya means inviting death upon one's self and although they are not scared of death but committing such deliberate suicide is not going to serve any good for anyone rather they will be tagged as idiots not warriors anymore. Hence they had to be together on the way.
Karna decided not to talk anymore and engaged himself in works. He had taken maximum responsibility upto himself , cutting snow with axe, putting tents, cutting woods and making fires, arranging foods everything. That atleast save him from losing patience on Arjun's sarcastic taunts, he very well knows once he loses patience he can't stop himself from attacking Arjun and that may cause both of their end and Karna is not ready to give his life nowhere else than battlefield , its his promise to himself to die while fighting a great war. so karna went on silent mode.
Arjun was caught with problems now. Since Angraj is not talking anymore hence he can't be rude to Karna for doing all works, and honestly Arjun is never good with such arrangements. when he and madhav used to set for adventure he always had left thing on Madhav. and when arjun was in 12 years exile he was never alone, first in pilgrimages with sages, then with , uloopi, then chitrangada , then again wandering in pilgrimages , meditating , then in dwarka with subhadra and madhav he never was alone, and he was not needed to arrange anything for himself, if he needed anything his archery helped. but he can't use archery here so in subconscious he was quite relieved for karna taking all responsibilities though didn't acknowledge. but question was how to kill time now.
Nature lover arjun although very soon got lost in divine beauty of Himalaya, it didn't take a long time for him to put up with the situation , his mind got lightened up by mountain valley's wonder. while karna got busy with work arjun got busy in enjoying nature.
after 2-3 days they reached Himalaya they camped in a snowy place, karna went in nearby mountainous tribal's village to arrange some food . after he came back he didn't find arjun anywhere, that made him tensed remembering the condition of trip , he started searching for this crazy prince and after a while what he watched a chilling thrill spread in his blood . Arjun is playing with a baby white bear and he is so engrossed in his sport that is not noticing from a distance mother bear is noticing him on his back. Karna had heard how much ferocious this mountainous white bears are if Arjun doesn't move right now he will bid adieu the world in no time. He can't use weapon to any Himalayan species that will evoke Devi shakti's wrath so what to do?
"Arjun look at your back" Karna tried to warn the playing rival. But Arjun didn't seem to hear.
"Arjun , leave the child , mother is near" karna almost shouted but Arjun didn't care. and increasing karna's palpitation mother bear started proceeding towards Arjun.
Karna couldn't take it any more he jumped on Arjun and threw the kid bear towards its mother and before mother bear could turn her concentration to them with all his might Karna dragged Arjun towards their camp as if an enraged father dragging his mischievous son to his room.
"what the hell were you doing"Karna was so angry and tensed that he was even not in place to express his anger too
"why did you care?" Arjun asked normally as if nothing happened.
"i was compelled to , since the condition of the trip is coming back alive together and am in no mood to give up my life to mountain god because of your idiocy instead of your arrows." karna gasped after talking without a pause
"did you really have confused me with Vrishasena ? its hard to accept for you that even i also possess something called warrior instinct.. you could at least wait till the mother bear reaches to me? i dealt with more wild species than you , forgot i was born in a forest? why did you get so hyper before nothing happened?"
Karna had no answers for Arjun's questions , so he silently left for finishing his rest of the work . Arjun grinned mischievously he has got some entertainment in the trip , teasing Karna by being more adventurous , quite unique idea to annoy one's rival.