_Notation.dox

1/*! \page Notation Notation for equations

2

3\section Symbols Symbols for variables

4

5\f$z\f$ refers to geopotential elevation (or height), increasing

6upward and with \f$z=0\f$ defining the resting ocean surface. Much of

7the ocean has \f$z < 0\f$.

8

9\f$x\f$ and \f$y\f$ are the Cartesian horizontal coordinates. MOM6

10 uses generalized orthogonal curvilinear horizontal

11 coordinates. However, the equations are simpler to write using

12 Cartesian coordinates, and it is very straightforward to generalize

13 the horizontal coordinates using the methods in Chapters 20 and 21 of

14 \cite SMGbook.

15

16\f$\lambda\f$ and \f$\phi\f$ are the geographic coordinates on a

17sphere (longitude and latitude, respectively).

18

19Horizontal components of velocity are indicated by \f$u\f$ and \f$v\f$

20and vertical component by \f$w\f$.

21

22\f$p\f$ is the hydrostatic pressure.

23

24\f$\Phi\f$ is the geopotential. In the absence of tides, the

25geopotential is given by \f$\Phi = g z,\f$ whereas more general

26expressions hold when including astronomical tide forcing.

27

28The thermodynamic state variables can be salinity, \f$S\f$, and

29potential temperature, \f$\theta\f$. Alternatively, one can choose

30the Conservative Temperature if using the TEOS10 equation of state

31from \cite TEOS2010.

32

33\f$\rho\f$ is the in-situ density computed as a function

34\f$\rho(S,\theta,p)\f$ for non-Boussinesq ocean or

35\f$\rho(S,\theta,p=-g \, \rho_o \, z)\f$ for Boussinesq ocean. See

36Young (2010) \cite Young2010 or Section 2.4 of Vallis (2017)

37\cite GVbook for reasoning behind the simplified pressure

38used in the Boussinesq equation of state.

39

40

41

42\section vector_notation Vector notation

43

44The three-dimensional velocity vector is denoted \f$\mathbf{v}\f$

45and it is decomposed into its horizontal and vertical components according to

46\f{align}

47\mathbf{v}

48 = \mathbf{u} + \hat{\mathbf{z}} \, w

49 = \hat{\mathbf{x}} \, u + \hat{\mathbf{y}} \, v + \hat{\mathbf{z}} \, w,

50 \f}

51where \f$\hat{\mathbf{z}}\f$ is the unit vector pointed in the

52upward vertical direction and \f$\mathbf{u} = (u, v, 0)\f$ is the

53horizontal component of velocity normal to the vertical.

54

55The three-dimensional gradient operator is denoted \f$\nabla\f$, and it is decomposed into

56its horizontal and vertical components according to

57\f{align}

58\nabla

59 = \nabla_z + \hat{\mathbf{z}} \, \partial_z

60 = \hat{\mathbf{x}} \, \partial_x + \hat{\mathbf{y}} \, \partial_y + \hat{\mathbf{z}} \, \partial_z.

61 \f}

62

63*/