For the primitive equations of large-scale atmosphere and ocean dynamics, we study the problem of determining by means of a variational data assimilation algorithm initial conditions that generate strong solutions which minimize the distance to a given set of time-distributed observations. We suggest a modification of the adjoint algorithm whose novel elements is to use norms in the variational cost functional that reflects the \(H^1\)-regularity of strong solutions of the primitive equations. For such a cost functional, we prove the existence of minima and a first-order adjoint condition for strong solutions that provides the basis for computing these minima. We prove the local convergence of a gradient-based descent algorithm to optimal initial conditions using the second-order adjoint primitive equations. The algorithmic modifications due to the \(H^1\)-norms are straightforwardly to implement into a variational algorithm that employs the standard \(L^2\)-metrics.

对于大规模大气和海洋动力学的原始方程式，我们研究了借助变分数据同化算法确定初始条件的问题，该条件产生了强大的解，从而使与给定时间分布的观测值之间的距离最小化。我们建议对伴随算法的一种修改，其新颖元素是在反映\（H ^ 1 \）的变分成本函数中使用范数-原始方程强解的正则性。对于这种成本函数，我们证明了极小值的存在和强解的一阶伴随条件，从而为计算这些极小值提供了基础。我们证明了基于梯度的下降算法的局部收敛性，可以使用二阶伴随原始方程来优化初始条件。由\（H ^ 1 \）-范数引起的算法修改可直接实现为采用标准\（L ^ 2 \）-度量的变分算法。