In a large bowl, whisk together the vinegar, turmeric, and 2 tablespoons of the olive oil. Add the arugula and radicchio and toss. Season with salt and pepper, to taste, and divide the mixture among serving plates. Set aside.
Season the salmon with salt and pepper, to taste. In a large skillet over medium-high heat, warm the remaining olive oil. Add the salmon and cook until barely opaque throughout, 2-3 minutes per side. Arrange the salmon on top of the salads.
Return the skillet to medium heat, add the shallots and jalapeño, and cook, stirring occasionally, until tender, 2-3 minutes. Add the cherries and ginger and cook, stirring occasionally, until heated through.
Add the cherry juice, increase the heat to medium-high, and cook, scraping up any browned bits in the skillet, until the juice is almost entirely evaporated, 2-3 minutes. Season with salt and pepper, to taste. Spoon the cherry mixture over the salmon and serve.