Problems
167
limit the cascade of energy to small scales are appropriate in those fluid-dynamical
applications where there actually is a systematic transfer of kinetic energy from
large to small scale . Indeed, any accurate numerical approximation to the equations goveming such flows must replicate this down-scale energy transfer.
One natural approach to the elimination of nonlinear instability in systems that
support a down-scale energy cascade is through the parametrization of unresolved
turbulent dissipation. In high-Reynolds-number (nearly inviscid) flow, kinetic energy is ultimately transferred to very small scales before being converted to internal energy by viscous dissipation, yet the storage limitations of digital computers
do not allow most numerical simulations to be conducted with sufficient spatial
resolution to resolve all the small-scale eddies involved in this energy cascade.
Under such circumstances the kinetic energy transferred down-scale during the
numerical simulation will tend to accumulate in the smallest scales resolvable on
the numerical mesh, and it is generally necessary to remove this energy by some
type of scale-selective dissipation. The scale -selective dissipation constitutes a
parametrization of the influence of the unresolved eddies on the resolved-scale
flow and should be designed to represent the true behavior of the physical
as closely as possible. Regardless of the exact formulation of the energy removal
scheme, it will tend to stabilize the solution and prevent nonlinear instability.
Many fluid flows contain limited regions of active srnall-scale turbulence and
relatively larger patches of dynamically stable laminar flow. Since eddy diffusion will not be active outside the regions of parametrized turbulence, a scaleselective background dissipation, similar to Phillips 's (1959) technique of removing all wavelengths shorter than four grid intervals, is often required in order
to avoid nonlinear instability. This dissipation may be implicitly included in the
time -differencing or in an upwind-biased spatial difference, or it may be explicitly added to an otherwise nondamping method using formulae such as those discussed in Section 2.4.3 . Although it is not required for stability, a small amount of
background dissipation may also be incorporated in numerical approximations to
linear partial differential equations to damp those short-wavelength components
of the numerical solution whose phase speed and group velocity are most seriously in error.
Problems
1. Verify that the leapfrog time-differenced shallow-water equations (3.14)
and (3.15) support a computational mode, and that the forward-backwarddifferenced system (3.17) and (3.18) does not, by solving their respective
discrete-dispersion relations for w.
2. Eliminate h from the finite-difference equations for the leapfrog unstaggered scheme (3.14) and (3.15) and compare the resulting higher-order
finite-difference approximation to the second-order PDE
167
limit the cascade of energy to small scales are appropriate in those fluid-dynamical
applications where there actually is a systematic transfer of kinetic energy from
large to small scale . Indeed, any accurate numerical approximation to the equations goveming such flows must replicate this down-scale energy transfer.
One natural approach to the elimination of nonlinear instability in systems that
support a down-scale energy cascade is through the parametrization of unresolved
turbulent dissipation. In high-Reynolds-number (nearly inviscid) flow, kinetic energy is ultimately transferred to very small scales before being converted to internal energy by viscous dissipation, yet the storage limitations of digital computers
do not allow most numerical simulations to be conducted with sufficient spatial
resolution to resolve all the small-scale eddies involved in this energy cascade.
Under such circumstances the kinetic energy transferred down-scale during the
numerical simulation will tend to accumulate in the smallest scales resolvable on
the numerical mesh, and it is generally necessary to remove this energy by some
type of scale-selective dissipation. The scale -selective dissipation constitutes a
parametrization of the influence of the unresolved eddies on the resolved-scale
flow and should be designed to represent the true behavior of the physical
as closely as possible. Regardless of the exact formulation of the energy removal
scheme, it will tend to stabilize the solution and prevent nonlinear instability.
Many fluid flows contain limited regions of active srnall-scale turbulence and
relatively larger patches of dynamically stable laminar flow. Since eddy diffusion will not be active outside the regions of parametrized turbulence, a scaleselective background dissipation, similar to Phillips 's (1959) technique of removing all wavelengths shorter than four grid intervals, is often required in order
to avoid nonlinear instability. This dissipation may be implicitly included in the
time -differencing or in an upwind-biased spatial difference, or it may be explicitly added to an otherwise nondamping method using formulae such as those discussed in Section 2.4.3 . Although it is not required for stability, a small amount of
background dissipation may also be incorporated in numerical approximations to
linear partial differential equations to damp those short-wavelength components
of the numerical solution whose phase speed and group velocity are most seriously in error.
Problems
1. Verify that the leapfrog time-differenced shallow-water equations (3.14)
and (3.15) support a computational mode, and that the forward-backwarddifferenced system (3.17) and (3.18) does not, by solving their respective
discrete-dispersion relations for w.
2. Eliminate h from the finite-difference equations for the leapfrog unstaggered scheme (3.14) and (3.15) and compare the resulting higher-order
finite-difference approximation to the second-order PDE