At 9th barbarian level, you gain a storm aspect which can be invoked, having base damage and a special effect, and who's mastery also grants you additional permanent boon.
Aspects:
Fire aspect
base damage - 1d10 fire
special effect - on fire
boon - You gain resistance to fire damage.
Lightning aspect
base damage - 1d8 lightning damage
special effect - stunning for 1 round.
boon - Other lightning damage you deal is increased by 1 dice size category.
Thunder aspect
base damage - Your invocation effects deal 1d6 sonic damage
special effect - Deal an extra 2d6 sonic damage
Boon - Your shout and breath attacks and abilities have their range increased by 10ft, and their damage increased by 1 dice size category.
Blizzard aspect
base damage - 1d10 cold damage
special effect - Targets take 1 damage to fatigue. If applied through a critically failed save, freezes solid for 1d3 rounds.
boon - You gain resistance to cold damage. You ignore difficult terrain while under the effects of rage.
At 9th barbarian level, learn 2 ways to invoke a storm aspect:
Weapon invocation - Spend 1 rest dice or 1 round of rage to raise a melee weapon and enchant it for 1 hour. For the duration, its attacks deal storm aspect base damage as extra damage.
Breath invocation - If you have a breath attack, increase its range by 5ft and cause it to deal storm aspect base damage as extra damage. If you do not, you can spend 1 rest dice, battle dice, or 1 round of rage to breathe a 20ft cone as a bonus action. It that deals 3x base damage on a failed dexterity save, and half on a success. On failure, applies special effect. This effect has a 1d4 < 2 cooldown dice after use.
Shout invocation - As an action, shout a chant of storms in a 15ft ray. The range is increased by 5ft per constitution bonus (min 1.) Creatures hit take 3x base damage + your constitution bonus on a failed strength save and have the special effect applied to them.
Sage gesture invocation - As a bonus action, deal base damage + your wisdom bonus at a target exposed tile within 20ft. If a creature is present, it must make a constitution save to halve damage, also applying the special effect on a failure.
Soulstorm invocation - You can spend 1 rest dice during rage to invoke the effect in a 25ft radius around yourself, dealing your storm aspects base damage + your constitution bonus to creatures that fail a dexterity save, and half on a success. Creatures that critically fail the save also have the special condition applied to them.
War invocation - You can spend 1 round of rage or 1 battle dice as a free action to immediately make a melee attack, either unarmed or with a melee weapon, dealing an additional 2x base damage as extra damage, adding battle dice to damage dealt if battle dice were spent, and applying the special condition on a critical hit. You can do so once per round.
Ability DCs are 12 + con + prof bonus.
At 10th barbarian level, learn 3 spells of any class, so long as they either deal the base damage type of your chosen storm aspect, or apply its special condition. They can be either cantrips, or spells up to lvl 4.
You may cast any spell you know that deals your storm aspects base damage type or applies its special condition by spending a number of rest dice equal to the spells lvl. If cast during a storm, the rest dice cost is reduced by 1.
At 10th barbarian level, choose once ability:
Powerful invocations - Your storm aspect invocations deal an extra dice of damage.
Greater storm magic - Learn an extra spell or cantrip for Storm magic.
Stormcaller - The rest dice cost of your barbarian abilities and spells you cast is reduced by 1 (min 0) when exposed under a storm.
At 11th barbarian level, choose 3 upgrades from the list below based on which storm aspect invocations you know.
Weapon invocation
Greater weapon invocation - The weapon invocation allows the weapon to deal an extra dice of damage.
Meteor crush - The weapon invocation allows the wielder to spend 1 rest dice when making a crush maneuver, dealing an extra x2 aspect base damage to target. Causes the attack to hit a 10ft cone instead.
Enchanted stormblade - The weapon invocation allows you to point it at a creature as an action, forcing them to make a wisdom save. On a failure, the aspects special condition is applied.
Breath invocation
Breath resonance art - Any cooldown chance dice associated with breath weapons are improved by 1 dice size category.
Dragonsage method - You can double the range of your breath attacks, but reduce them to a narrow cone. Choose an extra damage type: fire, cold, or lightning. Apply 1d6 of that damage to all breath attacks you make.
Storm breath mastery - The range of your breath attacks is increased by 5ft. Your storm aspect base damage is increased by 1 dice when applied or delivered by breath attacks. The DC of your breath attacks is increased by 1.
Shout invocation
Thrown shout - The range of your shout invocation is doubled.
True voice method - You cannot be muted or silenced. Any verbal component spell you cast has its level increased by 1.
Voice over reality - Your shout invocation requires two more saves, and applies the storm aspects special effect on a failure for both.
Sage gesture invocation
Sage gesture mastery - Increase the range of sage gesture mastery and all spells you cast that have a somatic component by 20ft. Double the area of effect of spells(if they have one) you cast that have somatic components, and sage gesture mastery now affects a 2x2 tile area.
Summon storm totem - When using sage gesture invocation, you can spend 1 rest dice or 1 battle dice to instead create a totem of the storms element (a localized blizzard, a standing pillar of fire, a minor whirlwind, or a perpetual lightning bolt, subject to GM) in the invocations target area for 1 minute. It deals 6x storm aspect base damage to creatures that start their turn in or enter the area.
Storm mark - When using sage gesture, you can spend 1 rest dice or 1 battle dice, or take 1 damage to fatigue, to instead affect a number of tiles equal to 2 + 2x your dexterity bonus within range. The tiles become infused with your storm aspect. Creatures that enter or start their turn on the tile must make a constitution save, taking 1d8 damage of the aspects base damage type and having the storm aspects special effect applied on a failure.
Soulstorm invocation
Eye of the storm - Increase range to 50ft. Allows leaving inner 5 to 10ft of effect untouched.
Stormrage - While raging, all your attacks deal an extra 1d8 damage of your storm aspects base damage type.
Soulstorm mastery - You can ignore up to 6 tiles within the soulstorm invocation, sparing creatures and objects within them. Soulstorm deals an extra dice of damage and applies the special effect on a failure.
War invocation
Greater war invocation - When you slay a creature with a war invocation attack, you can restore either 1 rest dice or 1 battle dice.
Storm striker - You can use war invocation twice per round. Your max battle dice are increased by 1.
Warrior of the storm - You gain a +1 bonus to AC and attacks while fighting under a storm.
At 12th barbarian level, you learn to call a storm. Once per long rest, you can speak the storm into being as a full round action. It lasts 2d4 / 2 hours.
If you have already used this ability since the last long rest, you can attempt to do it again, but doing so requires spending 4 rest dice.
During such a called storm, your aspect storm aspect and invocation damage is increased by 1d4(it is applied to all things that benefit from your aspects base damage.)
Depending on your storm aspect, you gain one of the following effects during such a called storm:
Fire aspect, Call comet - As an action, spend 4 rest dice and designate target 3x3 tile area within sight. At the start of the next round, all creatures within the area must make a DC 20 dexterity save. On a failure, they take 4d8 force damage, 4d8 fire damage, and 4d8 radiant damage. This has a 1d12 > 11 cooldown chance dice after use. On a failure they are allowed to relocate to a tile at least 10ft away from the affected tiles.
Lightning aspect, Call lightning bolt - As an action, spend 2 rest dice to either strike a lightning bolt at a target point within 80ft. Creatures within 5ft of target point must make a DC 20 wisdom or constitution save, taking 5d10 lightning damage on a failure and half on a success. On a failure, they are also stunned for 1 round, and are stunned for 1d3 rounds on a critical failure.
Thunder aspect, True storm speaking - The range of any spell you know becomes any point you can see under the storm. The range of all your ranged abilities is doubled while exposed to the storm. All sonic damage you deal is increased by 2d6.
Blizzard aspect, Direfrost caller - You are immune to cold damage. Any time you would stun a creature, they are instead frozen solid for the duration of the storm. Any negative condition you apply also hobbles creatures affected. You may choose to spend 1 rest dice when a creature dies under the storm, allowing them to become frozen solid instead, and granting them one more death save when they thaw(subject to GM.)