We propose a two-layer model with two different axes of integration and a well-balanced finite volume method. The purpose is to study submarine avalanches and generated tsunamis by a depth-averaged model with different averaged directions for the fluid and the granular layers. Two-layer shallow depth-averaged models usually consider either Cartesian or local coordinates for both layers. However, the motion characteristics of the granular layer and the water wave are different: the granular flow velocity is mainly oriented downslope while water motion related to tsunami wave propagation is mostly horizontal. As a result, the shallow approximation and depth-averaging have to be imposed (i) in the direction normal to the topography for the granular flow and (ii) in the vertical direction for the water layer. To deal with this problem, we define a reference plane related to topography variations and use the associated local coordinates to derive the granular layer equations whereas Cartesian coordinates are used for the fluid layer. Depth-averaging is done orthogonally to that reference plane for the granular layer equations and in the vertical direction for the fluid layer equations. Then, a finite volume method is defined based on an extension of the hydrostatic reconstruction. The proposed method is exactly well-balanced for two kind of stationary solutions: the classical one, when both water and granular masses are at rest; the second one, when only the granular mass is at rest. Several tests are presented to get insight into the sensitivity of the granular flow, deposit and generated water waves to the choice of the coordinate systems. Our results show that even for moderate slopes (up to ${30}^{\circ }$), strong relative errors on the avalanche dynamics and deposit (up to 60%) and on the generated water waves (up to 120%) are made when using Cartesian coordinates for both layers instead of an appropriate local coordinate system as proposed here.

我们提出了一个具有两个不同积分轴的两层模型，以及一个均衡的有限体积方法。目的是通过深度平均模型研究海底雪崩和产生的海啸，该平均模型对流体层和颗粒层的平均方向不同。两层浅深度平均模型通常考虑两层的笛卡尔坐标或局部坐标。但是，颗粒层和水波的运动特征是不同的：颗粒流速主要是向下倾斜，而与海啸波传播有关的水运动大多是水平的。结果，对于颗粒流必须在（i）垂直于形貌的方向上进行浅层近似和深度平均，（ii）对于水层必须在垂直方向上进行浅层近似和深度平均。为了解决这个问题，我们定义了与地形变化相关的参考平面，并使用相关的局部坐标来导出颗粒层方程，而笛卡尔坐标用于流体层。对于颗粒层方程式，垂直于该参考平面进行深度平均；对于流体层方程式，在垂直方向上进行深度平均。然后，基于静水力重建的扩展定义了有限体积法。所提出的方法对于两种固定解是完全平衡的：经典的一种，当水和颗粒团都静止时；另一种是静态的。第二个是只有颗粒团静止时。提出了一些测试以深入了解颗粒流，沉积物和生成的水波对坐标系选择的敏感性。${30}^{\circ }$），当对两层都使用笛卡尔坐标而不是此处建议的适当局部坐标系时，雪崩动力学和沉积物（高达60％）和生成的水波（高达120％）上会产生较大的相对误差。