_Tracer_Transport.dox

1/*! \page Tracer_Transport_Equations Tracer Transport Equations

2

3\image html PPM_1d.png "The 1-D finite volume advection of tracers. The reddish fluid will be in the cell at the end of the timestep."

4\image latex PPM_1d.png "The 1-D finite volume advection of tracers. The reddish fluid will be in the cell at the end of the timestep."

5

6Given a piecewise polynomial description of the tracer concentration, the new tracer cell

7concentration is the average of the fluid that will be in the cell after a timestep.

8

9\f{eqnarray}

10 \int_{x_{i-1/2}}^{x_{i+1/2}} A_i^{n+1} (x) dx =

11 \int_{x_{i-1/2 - u \Delta t}}^{x_{i+1/2-u\Delta t}} A_i^{n} (x) dx &= \mbox{} \\

12 \int_{x_{i-1/2}}^{x_{i+1/2}} A_i^{n} (x) dx -

13 \int_{x_{i+1/2 - u \Delta t}}^{x_{i+1/2}} A_i^{n} (x) dx &+

14 \int_{x_{i-1/2 - u \Delta t}}^{x_{i-1/2}} A_i^{n} (x) dx

15\f}

16

17Fluxes are found by analytically integrating the profile over the distance that is

18swept past the face within a timestep.

19

20\f[

21 a_i^n = \frac{1}{\Delta x} \int_{x_{i-1/2}}^{x_{i+1/2}} A_i^n(x) dx

22\f]

23\f[

24 a_i^{n+1} = a_i^n - \frac{\Delta t}{\Delta x} (F_{i+1/2} - F_{i-1/2})

25\f]

26\f[

27 F_{i+1/2} = \frac{1}{\Delta t} \int_{x_{i+1/2 - u \Delta t}}^{x_{i+1/2}} A_i^n(x) dx

28\f]

29\f[

30 F_{i-1/2} = \frac{1}{\Delta t} \int_{x_{i-1/2 - u \Delta t}}^{x_{i-1/2}} A_i^n(x) dx

31\f]

32

33With piecewise constant profiles, this approach give first order upwind advection.

34Higher order polynomials (e.g., parabolas) can give higher order accuracy.

35

36\section Multidimensional_Tracer_Advection Multidimensional Tracer Advection

37

38Using "Easter's Pseudo-compressibility" (\cite easter1993), we start with these

39basic equations for a tracer \f$\psi\f$:

40

41\anchor ht-equation

42\f[

43 \frac{\partial h}{\partial t} + \vec{\nabla} \cdot (\vec{u}h) = 0 \equiv

44 \frac{\partial h}{\partial t} + \vec{\nabla} \cdot (\vec{U})

45\f]

46

47\f[

48 \frac{\partial}{\partial t} (h \psi) + \vec{\nabla} \cdot (\vec{U}\psi) = 0

49\f]

50

51\f[

52 \frac{\partial \psi}{\partial t} + \vec{u} \cdot \vec{\nabla} \psi = 0

53\f]

54

55We discretize the first of these equations in space:

56

57\f[

58 \frac{\partial h}{\partial t} = \frac{1}{\Delta x} \left(U_{i-\frac{1}{2},j} -

59 U_{i+\frac{1}{2},j} \right) + \frac{1}{\Delta y} \left(V_{i, j-\frac{1}{2}} -

60 V_{i,j+\frac{1}{2}} \right)

61\f]

62

63Using our monotonic one-dimensional flux:

64

65\f[

66 F_{i+\frac{1}{2},j} (\psi) = U_{i+\frac{1}{2},j} \psi_{i+\frac{1}{2},j}

67\f]

68

69we come up with an estimate based only on an update in the \f$x\f$ direction:

70

71\f[

72 \tilde{h}_{i,j} \tilde{\psi}_{i,j} = h^n_{i,j} \psi_{i,j} + \frac{\Delta

73 t}{\Delta x} \left( F_{i-\frac{1}{2},j} (\psi^n) - F_{i+\frac{1}{2},j} (\psi^n)

74 \right)

75\f]

76

77\f[

78 \tilde{h}_{i,j} = h^n_{i,j} + \frac{\Delta

79 t}{\Delta x} \left( U_{i-\frac{1}{2},j} - U_{i+\frac{1}{2},j} \right)

80\f]

81

82\f[

83 \tilde{\psi}_{i,j} = \frac{\tilde{h}_{i,j} \tilde{\psi}_{i,j}}{\tilde{h}_{i,j}}

84\f]

85

86Next, we update in the \f$y\f$ direction:

87

88\f[

89 h^{n+1}_{i,j} \psi^{n+1}_{i,j} = \tilde{h}_{i,j} \tilde{\psi}_{i,j} + \frac{\Delta

90 t}{\Delta y} \left( G_{i,j-\frac{1}{2}} (\tilde{\psi}) - G_{i,j+\frac{1}{2}}

91 (\tilde{\psi}) \right)

92\f]

93

94\f[

95 h^{n+1}_{i,j} = \tilde{h}_{i,j} + \frac{\Delta

96 t}{\Delta y} \left( V_{i,j-\frac{1}{2}} - V_{i,j+\frac{1}{2}} \right)

97\f]

98

99\f[

100 \psi^{n+1}_{i,j} = \frac{h^{n+1}_{i,j} \psi^{n+1}_{i,j}}{h^{n+1}_{i,j}}

101\f]

102

103\li This method ensures monotonicity. Strang splitting can reduce directional

104splitting error. See \cite easter1993, \cite durran2010 (section 5.9.4), and

105\cite russell1981 .

106

107\li Flux-form pseudo-compressibility advection is based on accumulated mass (or

108volume) fluxes, not velocities.

109

110\li Additional pseudo-compressibility passes can be added to accommodate

111transports exceeding cell masses. Extra passes of tracer advection are used in

112MOM6 in the small fraction of cells where this is needed.

113

114\li Explicit layered dynamics time-steps are limited by Doppler-shifted internal

115gravity wave speeds or inertial oscillations.

116Flow speeds in most of the ocean volume are much smaller than the peak

117internal wave speeds so that the advective time-steps can be longer.

118

119\li Advective mass fluxes in MOM6 are often accumulated over multiple dynamic

120steps. The goal is that as we go to higher resolution, this tracer advection will

121remain stable at relatively long time-steps, allowing for the inclusion of many

122biogeochemical tracers without adding an undue burden in computational cost.

123

124*/