Theoretical Background

Since the ”mixing length” theory has been widely used to analyze turbulence in geo- physical and atmospheric flows, many models has been proposed to give the variation of the ”mixing length” with height in the ABL (Prandtl, 1932; Blackadar, 1962; Panofsky, 1973; Lettau, 1962; Gryning et al., 2007), which leads to our current understanding of

turbulent mixing in the ABL. Many of these models are derived based on the observed mean wind profile and the turbulent mixing equation which relates the mixing length to the mean wind profile, since the mixing length itself is hard, if not impossible, to be measured directly. As the dropwindsonde gives an unprecedented opportunity to cal- culate vertical turbulence correlation coefficients in hurricanes, the turbulence integral length scale can be directly integrated from these correlation coefficients and compared to the mixing length inferred from the mean profile following the methodology given by Panofsky (1973) and Pena et al. (2010). Such a comparison may not only advance our understanding of turbulent mixing in the HBL but amy also provide additional evi- dence supporting the relationship between the turbulent integral length scale and mixing length.

Meanwhile, the PBL scheme has been widely employed in current numerical simulation packages concerning atmospheric flows (Nolan et al., 2009b,a). Non-local PBL schemes, such as the YSU scheme used in WRF, prescribe and parametrize the turbulence diffu- sivity profile, which is critical in the overall atmospheric flow simulation since it reflects the influence of the surface boundary condition and modifies the fields predicted in the lower portion of the atmosphere. These PBL schemes are, however, mainly designed to describe turbulent mixing in the standard ABL (Hong and Pan, 1996; Hong et al., 2006). Thus, its validity in simulating turbulent mixing in the HBL is questionable despite the fact that they are already widely adopted in researches and the numerical forecasts of hurricanes. It is worth comparing the turbulence length scale derived from these PBL schemes to the turbulent integral length scale calculated based on dropwindsonde mea- surements, which not only checks the validity of such PBL schemes in simulating the HBL turbulence but also provides opportunities to make suggestions on possible improvements upon its original formulation.

Following and extending the study of Pena et al. (2010), the turbulence diffusivity can be expressed as the product of a turbulent velocity scale and a length scale following the Prandlt’s mixing length theory. As in the first-order closure, the momentum flux

τ =ρu′_w′ _{for the longitudinal and vertical components of the wind can be related to the} derivative of the mean wind velocity as,

τ =ρKmdU

dz (4.3)

where ρ gives the density of the air, U is the longitudinal wind component and the

turbulence diffusivity for the momentum, Km, can be expressed as

Km =l2 dU dz (4.4) In equation (4.4),lrepresents the mixing length. Once the momentum flux τ is modelled based on the shear velocity u∗ as,

τ =ρu2_∗ (4.5)

the turbulence diffusivity of momentum,Km, can be derived, combining expressions (4.4) and (4.5), as a function of the shear velocity u_∗ and mixing length l as,

Km =u∗(z)l(z) (4.6)

It should be noted that this shear velocity u∗ does not only reflect the shear stress at surface, but is a function of height,u_∗(z), giving the shear stress for the entire boundary layer. Following Panofsky (1973); Gryning et al. (2007); Pena et al. (2010), the vertical variation of the shear stress in the entire boundary layer can be expressed in terms of the shear velocity as,

u∗(z) =u∗0(1−

z

zi) (4.7)

wherezi gives the boundary layer height and u∗0 gives theu∗ value at surface. Combing expressions (4.7) and (4.6), the turbulence diffusivity for the momentum can be rewritten as,

Km =u_∗0(1−

z

zi)l(z) (4.8)

where the mixing lengthl should also be a function of height, and therefore rewritten as

l(z).

Although many models using a mixing length profile l(z) have been proposed, as

summarized by Pena et al. (2010), only the length model implicitly used by the turbulence diffusivity formulation of the YSU scheme is investigated here. As described by Hong and Pan (1996) and Hong et al. (2006), the turbulence diffusivity profile of the YSU scheme, based on the study of Brost and Wyngaard (1978) and Troen and Mahrt (1986), is prescribed as a function of height as,

Km =kwsz(1− z zi)

p _(4.9)

where k is the von-Karman constant, ws = u∗0φ−1 is the mixed layer velocity, and p

is the model parameter that currently takes the value of 2.0 in the operational WRF. Rewriting it in the same format as in equation (4.8), Km used by the YSU scheme is,

Km =u∗0φ−1(1− z zi)·kz(1− z zi) p−1 _(4.10)

In comparing equation (4.10) to equation (4.8), it can be seen that the turbulent velocity

scale used in YSU scheme has one more parameter φ−1_{, which is the Monin-Obukhov}

function evaluated at the top of the surface layer, and the mixing length, implied by the YSU scheme, is modelled as,

l(z) =kz(1− z zi

)p−1 _(4.11)

Given that the model parameter ptakes the value of 2.0, equation (4.11) shows that the YSU scheme requires the turbulence length scale, or the mixing length, to be a parabolic function of height.