Turbulent kinetic energy equation
The TurbulentKineticEnergyEquation module provides diagnostics for the turbulent kinetic energy (TKE) budget. While the KineticEnergyEquation module deals with the total (resolved) kinetic energy, this module focuses on the kinetic energy of velocity perturbations relative to a specified mean flow. TKE is defined as
\[e = \tfrac{1}{2} u_i' u_i'\]
where $u_i' = u_i - U_i$ is the velocity perturbation from the mean $U_i$. Mean velocities default to zero (in which case TKE reduces to KE) and can be set via the U, V, W keyword arguments.
A key term in the TKE budget is the shear production rate, which quantifies the transfer of kinetic energy from the mean flow to the turbulence through Reynolds stresses acting on the mean shear:
\[P = -u_i' u_j' \partial_j U_i\]
The module provides both directional components ($P_x$, $P_y$, $P_z$) and the total shear production. It also provides an isotropic dissipation rate diagnostic that computes $\varepsilon = 2\nu S'_{ij} S'_{ij}$ using the perturbation strain rate tensor.
All diagnostics are computed at (Center, Center, Center).
Example
julia> using Oceananigans, Oceanostics
julia> grid = RectilinearGrid(size=(4, 4, 4), extent=(1, 1, 1));
julia> model = NonhydrostaticModel(grid; closure=ScalarDiffusivity(ν=1e-4));
julia> tke = TurbulentKineticEnergyEquation.TurbulentKineticEnergy(model)
TurbulentKineticEnergy KernelFunctionOperation at (Center, Center, Center)
├── grid: 4×4×4 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── kernel_function: turbulent_kinetic_energy_ccc (generic function with 1 method)
└── arguments: ("Field", "Field", "Field", "Oceananigans.Fields.ZeroField", "Oceananigans.Fields.ZeroField", "Oceananigans.Fields.ZeroField")
└── computes: turbulent kinetic energy ½uᵢ′uᵢ′
julia> SP = TurbulentKineticEnergyEquation.ShearProductionRate(model)
TurbulentKineticEnergyShearProductionRate KernelFunctionOperation at (Center, Center, Center)
├── grid: 4×4×4 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── kernel_function: shear_production_rate_ccc (generic function with 1 method)
└── arguments: ("Field", "Field", "Field", "Oceananigans.Fields.ZeroField", "Oceananigans.Fields.ZeroField", "Oceananigans.Fields.ZeroField")
└── computes: total TKE shear production -uᵢ′uⱼ′ ∂ⱼUᵢTurbulent kinetic energy
Oceanostics.TurbulentKineticEnergyEquation.TurbulentKineticEnergy — Type
TurbulentKineticEnergy(model, u, v, w; U, V, W, location)
Calculate the turbulent kinetic energy of model.
julia> using Oceananigans, Oceanostics
julia> grid = RectilinearGrid(size=(4, 4, 4), extent=(1, 1, 1));
julia> model = NonhydrostaticModel(grid);
julia> TKE = TurbulentKineticEnergyEquation.TurbulentKineticEnergy(model)
TurbulentKineticEnergy KernelFunctionOperation at (Center, Center, Center)
├── grid: 4×4×4 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── kernel_function: turbulent_kinetic_energy_ccc (generic function with 1 method)
└── arguments: ("Field", "Field", "Field", "Oceananigans.Fields.ZeroField", "Oceananigans.Fields.ZeroField", "Oceananigans.Fields.ZeroField")
└── computes: turbulent kinetic energy ½uᵢ′uᵢ′Isotropic dissipation rate
Oceanostics.TurbulentKineticEnergyEquation.TurbulentKineticEnergyIsotropicDissipationRate — Function
TurbulentKineticEnergyIsotropicDissipationRate(
u,
v,
w,
args;
U,
V,
W,
location
)
Calculate the Turbulent Kinetic Energy Isotropic Dissipation Rate, defined as
ε = 2 ν S'ᵢⱼS'ᵢⱼ,where S'ᵢⱼ is the strain rate tensor, for a fluid with an isotropic turbulence closure (i.e., a turbulence closure where ν (eddy or not) is the same for all directions.
julia> using Oceananigans, Oceanostics
julia> grid = RectilinearGrid(size=(4, 4, 4), extent=(1, 1, 1));
julia> model = NonhydrostaticModel(grid; closure=ScalarDiffusivity(ν=1e-4));
julia> TurbulentKineticEnergyEquation.IsotropicDissipationRate(model)
KineticEnergyIsotropicDissipationRate KernelFunctionOperation at (Center, Center, Center)
├── grid: 4×4×4 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── kernel_function: isotropic_viscous_dissipation_rate_ccc (generic function with 1 method)
└── arguments: ("Field", "Field", "Field", "NamedTuple")
└── computes: isotropic kinetic energy dissipation rate ε = 2νSᵢⱼSᵢⱼShear production rates
Oceanostics.TurbulentKineticEnergyEquation.TurbulentKineticEnergyXShearProductionRate — Type
TurbulentKineticEnergyXShearProductionRate(
u′,
v′,
w′,
U,
V,
W;
grid,
location
)
Calculate the shear production rate in the model's x direction:
XSHEAR = uᵢ′u′∂x(Uᵢ)where uᵢ′ is the velocity perturbation in the i direction, u′ is the velocity perturbation in the x direction, Uᵢ is the background velocity in the i direction, and ∂x is the horizontal derivative.
julia> using Oceananigans, Oceanostics
julia> grid = RectilinearGrid(size=(4, 4, 4), extent=(1, 1, 1));
julia> model = NonhydrostaticModel(grid);
julia> XSHEAR = TurbulentKineticEnergyEquation.XShearProductionRate(model)
TurbulentKineticEnergyXShearProductionRate KernelFunctionOperation at (Center, Center, Center)
├── grid: 4×4×4 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── kernel_function: shear_production_rate_x_ccc (generic function with 1 method)
└── arguments: ("Field", "Field", "Field", "Oceananigans.Fields.ZeroField", "Oceananigans.Fields.ZeroField", "Oceananigans.Fields.ZeroField")
└── computes: TKE shear production (x) -uᵢ′u′ ∂ₓUᵢOceanostics.TurbulentKineticEnergyEquation.TurbulentKineticEnergyYShearProductionRate — Type
TurbulentKineticEnergyYShearProductionRate(
u′,
v′,
w′,
U,
V,
W;
grid,
location
)
Calculate the shear production rate in the model's y direction:
YSHEAR = uᵢ′v′∂y(Uᵢ)where uᵢ′ is the velocity perturbation in the i direction, v′ is the velocity perturbation in the y direction, Uᵢ is the background velocity in the i direction, and ∂y is the vertical derivative.
julia> using Oceananigans, Oceanostics
julia> grid = RectilinearGrid(size=(4, 4, 4), extent=(1, 1, 1));
julia> model = NonhydrostaticModel(grid);
julia> YSHEAR = TurbulentKineticEnergyEquation.YShearProductionRate(model)
TurbulentKineticEnergyYShearProductionRate KernelFunctionOperation at (Center, Center, Center)
├── grid: 4×4×4 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── kernel_function: shear_production_rate_y_ccc (generic function with 1 method)
└── arguments: ("Field", "Field", "Field", "Oceananigans.Fields.ZeroField", "Oceananigans.Fields.ZeroField", "Oceananigans.Fields.ZeroField")
└── computes: TKE shear production (y) -uᵢ′v′ ∂_yUᵢOceanostics.TurbulentKineticEnergyEquation.TurbulentKineticEnergyZShearProductionRate — Type
TurbulentKineticEnergyZShearProductionRate(
u′,
v′,
w′,
U,
V,
W;
grid,
location
)
Calculate the shear production rate in the model's z direction:
ZSHEAR = uᵢ′w′∂z(Uᵢ)where uᵢ′ is the velocity perturbation in the i direction, w′ is the vertical velocity perturbation, Uᵢ is the background velocity in the i direction, and ∂z is the vertical derivative.
julia> using Oceananigans, Oceanostics
julia> grid = RectilinearGrid(size=(4, 4, 4), extent=(1, 1, 1));
julia> model = NonhydrostaticModel(grid);
julia> ZSHEAR = TurbulentKineticEnergyEquation.ZShearProductionRate(model)
TurbulentKineticEnergyZShearProductionRate KernelFunctionOperation at (Center, Center, Center)
├── grid: 4×4×4 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── kernel_function: shear_production_rate_z_ccc (generic function with 1 method)
└── arguments: ("Field", "Field", "Field", "Oceananigans.Fields.ZeroField", "Oceananigans.Fields.ZeroField", "Oceananigans.Fields.ZeroField")
└── computes: TKE shear production (z) -uᵢ′w′ ∂_zUᵢOceanostics.TurbulentKineticEnergyEquation.TurbulentKineticEnergyShearProductionRate — Type
TurbulentKineticEnergyShearProductionRate(
u′,
v′,
w′,
U,
V,
W;
grid,
location
)
Calculate the total shear production rate (sum of the shear production rates in the model's x, y and z directions):
SHEAR = XSHEAR + YSHEAR + ZSHEAR = uᵢ′uⱼ′∂ⱼ(Uᵢ)where XSHEAR, YSHEAR and ZSHEAR are the shear production rates in the x, y and z directions, respectively, uᵢ′ and uⱼ′ are the velocity perturbations in the i and j directions, respectively, and ∂ⱼ is the derivative in the j direction.
julia> using Oceananigans, Oceanostics
julia> grid = RectilinearGrid(size=(4, 4, 4), extent=(1, 1, 1));
julia> model = NonhydrostaticModel(grid);
julia> SHEAR = TurbulentKineticEnergyEquation.ShearProductionRate(model)
TurbulentKineticEnergyShearProductionRate KernelFunctionOperation at (Center, Center, Center)
├── grid: 4×4×4 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── kernel_function: shear_production_rate_ccc (generic function with 1 method)
└── arguments: ("Field", "Field", "Field", "Oceananigans.Fields.ZeroField", "Oceananigans.Fields.ZeroField", "Oceananigans.Fields.ZeroField")
└── computes: total TKE shear production -uᵢ′uⱼ′ ∂ⱼUᵢ