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dTKEdt.m
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dTKEdt.m
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function epsilon = dTKEdt(z, ustar, wstar, BLmixingDepth, L)
%
% Matt Werner ([email protected]) - July 1, 2021
%
% Calculate the dissipation rate of the wind's turbulent kinetic energy
% (TKE) within the atmosphere's convective boundary layer (CBL). This
% expression provides a profile "for which both buoyancy- and
% shear-generated turbulence are important" (Aylor, eq. 9.45).
% 3 -1/4
% 0.4 w* 3 / z \ (1 - 15 z/L)
% epsilon = ---------- + u* | 1 - ------ | ----------------- ,
% z_i \ z_i / k z
%
% where wstar is the convective velocity scale, ustar is the friction
% velocity, z_i is the boundary layer mixing depth, z is the altitude, and
% L is the Obukhov length. The parameter k here (= 0.4) is von Karman's
% constant.
%
% Inputs:
%
% z - Altitude.
% Size: 1-by-1 (scalar)
% Units: m (meters)
%
% ustar - Friction velocity.
% Size: 1-by-1 (scalar)
% Units: m/s (meters per second)
%
% wstar - Convective velocity scale.
% Size: 1-by-1 (scalar)
% Units: m/s (meters per second)
%
% BLmixingDepth - Boundary layer mixing depth, a key parameter.
% Size: 1-by-1 (scalar)
% Units: m (meters)
%
% L - Obukhov length.
% Size: 1-by-1 (scalar)
% Units: m (meters)
%
% Outputs:
%
% epsilon - The rate of dissipation of turbulent kinetic energy
% within the convective boundary layer (CBL).
% Size: 1-by-1 (scalar)
% Units: m2/s3 (square meters per cubic second)
%
%% Checks
% No checks
%% Computation
% Calculate the rate of dissipation of TKE
epsilon = 0.4*wstar^3/BLmixingDepth + ustar^3*(1 - z/BLmixingDepth)*(1 - 15*z/L)^-0.25/(0.4*z);