_Barotropic_Momentum.dox

1/*! \page Barotropic_Momentum_Equations Barotropic Momentum Equations

2

3\brief Barotropic Momentum Equations

4

5The barotropic equations are timestepped on a relatively short timestep compared to the

6rest of the model. Since the timestep constraints for this are known, the barotropic

7timestep is computed at runtime.

8

9The 2-d linear momentum equations with integrated continuity are:

10

11\f[

12 \frac{\partial \eta}{\partial t} + \nabla \cdot \left( ( D + \eta) \vec{u}_{BT}

13 h_k \right) = P - E

14\f]

15\f[

16 \frac{\partial \vec{u}_{BT}}{\partial t} = - g \nabla \eta - f \hat{z} \times

17 \vec{u}_{BT} + \vec{F}_{BT}

18\f]

19where

20\f[

21 \vec{u}_{BT} \equiv \frac{1}{D + \eta} \int_{-D}^\eta \vec{u}dz

22\f]

23

24and \f$\vec{F}_{BT}\f$ is the barotropic momentum forcing from baroclinic

25processes. Note that explicit mass fluxes such as evaporation and

26precipitation change the model volume explicitly.

27

28In the mode splitting between baroclinic and barotropic processes, it is important

29to include the contribution of free surface waves on the internal interface

30heights on the pressure gradient force, shown here as \f$g_{Eff}\f$:

31

32\f[

33 \frac{\partial p}{\partial z} = -\rho g

34\f]

35\f[

36 g_{Eff} = g + \frac{\partial}{\partial \eta} \left[ \frac{1}{D + \eta}

37 \int_{-D}^\eta p dz \right]

38\f]

39

40The barotropic momentum equation then becomes:

41

42\f[

43 \frac{\partial \vec{u}_{BT}}{\partial t} + f \hat{z} \times

44 \vec{u}_{BT} + \frac{1}{\rho_0} \nabla g_{Eff} \eta = \mbox{Residual}

45\f]

46

47Without including the internal wave motion in the barotropic equations, one can

48generate instabilities (\cite bleck1990, \cite hallberg1997a).

49

50*/