When Katara is asked to be the new Avatar's Waterbending master, she is honored. She would have agreed to do the proposition even if the girl had been born in the North Pole, though privately she is both relieved and proud when the new Avatar is found in one of the Southern Tribe homes; already a healthy, lively, and strong girl. Relieved because she won't be expected to leave her beloved home again. The North Pole would have been a close approximation, climate-wise at least, but the snows in the South Pole are different. Katara can tell.
And proud because she chooses to see the birth of the Avatar in the Southern Water Tribes as a sign that her country's spirit is finally recovered from the brutal genocide they sufffered at the hands of the previous Fire Lords (she knows that's not necessarily the case; that the Avatar just needed to be a Waterbender and could have been born anywhere, even at the Fire Islands themselves but she wasn't; she was a Southern tribeswoman and her people's confidence would be bolstered because of it). The spirits were smiling down at the Southern Water Tribe the day Korra was born.
She likes to pretend that Aang's spirit was smiling down at her the moment Korra was born.
It's a dangerous fantasy. She knows it is. How can she pretend to possess enough equanimity to instruct the new Avatar while she daydreams that the girl's a memento from her dead husband? The Avatar would already have the weight of the world on her young shoulders. Did Katara truly wish to burden her with the loneliness of an old woman yearning for someone whose death she should've already come to terms with? No, Katara would need to forget that Aang had been the previous Avatar in order to avoid dumping absurd expectations on Korra's young shoulders. It would be folly let herself become convinced that Korra and Aang were the same person no matter how similar they might turn out to be.
She doesn't change her mind about becoming the girl's Waterbending master. It would be cowardice. Korra deserved the best Waterbender in the world to be her instructor (Katara's conviction that she's the best Waterbender in the world ins't arrogance--it's self-awareness). She steels herself and gets ready to be the best possible master for the Korra.
At first, it's not difficult to separate Aang from Korra. Prodigious bending talent aside, they're not all that similar. Even physically, they have little in common. Aang's frame had been slight, his skin pale when compared to a Water Trribesman's, and his eyes a warm, rich shade of brown. Korra is bulky and muscular, her skin a smooth shade of Water Tribe-brown, and her eyes as blue as a cloudless sky.
Korra's also eager to begin her life as the Avatar, to bring even more harmony into the world, and to protect those too weak to protect themselves. And more than confident that she had the skills to do so despite her youth and naivete. Aang's opposite indeed. Katara loves her, and not just because she isn't Aang's ghost. Korra's unapologetic individuality and fearless independence (from Aang, from her parents, from Katara, from everyone and everything) sets Katara free.
Then the spring snows arrive and Korra begins to sneak away from her lessons (Boring, she complains, when can we stop with the repetive drills?) to play with the younger kids of the White Lotus guards. Katara knows what she's doing and reluctantly remembers Aang's continued avoidance of training even as Sozin's comet nipped at their heels.
It's not the same, she tells herself as she makes her way towards the frozen pond she knows Korra will be using to impress the young kids with fancy Waterbending tricks, willing herself to have patience. The world won't end if Korra takes a little longer to master Waterbending.
She finds the kids chasing after someone (probably Korra) who's covered themselves in snow. "Get it!" yells one of the girls, clumsily making Waterbending motions towards the snow at her feet. Not a single flake obeys her childish commands.
"The Abominable Snowman's coming for your tooooys!" yells Korra, shambling towards the kids with her arms extended forward.
The kids laugh and try to run. One of them rushes towards the pond and Katara starts to go forward, afraid that the child will sink into the pond's undoubtedly melting water. Korra gets to the boy before she does, gently raising the ice in front of the boy to stop him from going forward. As gently as Korra does anything, anyway. The kids notice Korra's sudden tension and abruptly stop on their tracks. Katara starts forward again, the beginnings of a lecture about the dangers of bringing children near a defrosting pond at the tip of her tongue, and Korra suddenly drops to the ground and lets the snow she'd molded to her body fall off her.
"The effort has defeated me!" she cries, voice full of exaggerated anguish. "You brave Southern Tribe warriors have defeated the evil snow spirit and the South Pole is once again safe!"
Her words lift what little fear had tensed the little kids' shoulders and made them burst with happy laughter. A little girl drops beside Korra and makes a slashing motion with her small arm. Korra curls into a fetal position, as though the child has delivered a killing blow on some fearsome beast, and then bursts into happy laughter herself. The remaining two children also drop beside her and begin to beg for another round of play. Korra starts babbling about another monster Katara's sure she's made up on the spot and gets up, motioning in the direction of the mansion.
Katara knows she ought to reveal herself and deliver a serious lecture about the dangers of playing in defrosting ponds without adult supervision, but she's wondering if her heart hasn't forgotten how to pump blood towards her limbs.
For the first time, Korra seems like Aang embodied in a Waterbending girl.
Katara watches them return to the mansion without moving away from the rock she is standing behind. She told herself to take a few minutes to compose herself. It will be necessary if she wants to speak to Korra about safety without bursting into despairing-happy tears.