The contribution of Michael J. P. Cullen was written in the course of his employment at the Met Office, UK and is published with the permission of the Controller of HMSO and the Queen's Printer for Scotland

Research Article

# A semi-geostrophic model incorporating well-mixed boundary layers

Article first published online: 5 MAY 2010

DOI: 10.1002/qj.612

Copyright © 2010 Royal Meteorological Society and Crown Copyright.

Issue

## Quarterly Journal of the Royal Meteorological Society

Volume 136, Issue 649, pages 906–917, April 2010 Part B

Additional Information

#### How to Cite

Beare, R. J. and Cullen, M. J. P. (2010), A semi-geostrophic model incorporating well-mixed boundary layers. Q.J.R. Meteorol. Soc., 136: 906–917. doi: 10.1002/qj.612

#### Publication History

- Issue published online: 14 JUN 2010
- Article first published online: 5 MAY 2010
- Manuscript Accepted: 25 FEB 2010
- Manuscript Revised: 25 JAN 2010
- Manuscript Received: 15 OCT 2009

- Abstract
- Article
- References
- Cited By

### Keywords:

- semi-geotriptic;
- hydrostatic primitive;
- sea breeze;
- diurnal cycle;
- physics– dynamics coupling

### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. The primitive and semi-geotriptic models
- 3. Parametrizations and numerical methods
- 4. Results
- 5. Discussion
- References

Semi-geostrophic theory has proved a powerful framework for understanding the dynamics of mid-latitude weather systems. However, one limitation is the lack of a realistic boundary-layer representation. Semi-geostrophic theory can be modified to include an atmospheric boundary layer by replacing the geostrophic wind with the ‘geotriptic’ (or Ekman-balanced) value in the substantive derivative and appropriately approximating the momentum diffusion term– the so-called semi-geotriptic theory. However, until now, solutions of the semi-geotriptic equations using predictor– corrector methods have not been possible for the important case of well-mixed boundary layers. Existing predictor– corrector methods require a Brunt– Väisälä frequency greater than zero to be solvable.

Here we describe a method of incorporating well-mixed boundary layers into semi-geotriptic theory. We modify the hydrostatic relationship by including a small horizontal diffusion of vertical velocity. This enables the formation of a well-posed predictor– corrector method. Given well-mixed boundary layers are a ubiquitous feature of the lower atmosphere, the modification increases the usability of the model. Calculations are also performed at much higher vertical resolution than before.

The revised semi-geotriptic model is compared with a hydrostatic primitive-equation model for a test case of a two-dimensional idealized diurnal cycle of a sea breeze. The performance of the revised semi-geotriptic model in the growth phase of the sea breeze is improved, as a well-mixed boundary layer is now permitted. The additional vertical resolution captures the capping inversion and the sea-breeze circulation better. The hydrostatic primitive-equation model is shown to produce inertial oscillations that persist beyond the evening decay of the boundary layer until the following morning. In contrast, the semi-geotriptic model decays following the boundary-layer state in a more realistic way. The semi-geotriptic model thus demonstrates its usefulness as a critical tool in understanding boundary-layer dynamics coupling issues in operational models. Copyright © 2010 Royal Meteorological Society and Crown Copyright.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. The primitive and semi-geotriptic models
- 3. Parametrizations and numerical methods
- 4. Results
- 5. Discussion
- References

Since the seminal work of Hoskins and Bretherton (1972) and Hoskins (1975), the semi-geostrophic (SG) equations have proved a powerful framework for understanding the dynamics of mid-latitude weather systems. The only difference between the SG and hydrostatic primitive (HP) equations is the geostrophic momentum approximation; here the Lagrangian *advected* momentum is assumed geostrophic. As such, the SG equations have greater applicability than the quasi-geostrophic equations, which also restrict the *advecting* momentum to being geostrophic and the basic state stratification to being a function of height only.

One limitation of the SG equations for interpreting and understanding the behaviour of weather and climate models is that they do not include a realistic representation of the boundary layer. Although some authors have included the boundary layer using either a simplified Ekman pumping at the bottom boundary or a constant momentum diffusivity (Wu and Blumen 1982), there is very little in the literature including a realistic boundary-layer parametrization scheme for both momentum and heat. In order to include realistic boundary-layer diffusivity in the SG equations, Cullen (1989, 2006) introduced the semi-*geotriptic* (SGT) equations, where the geostrophic momentum is replaced by its ‘geotriptic’ (Earth ‘rubbing’) value and a full vertical diffusivity used for the boundary layer; the geotriptic (or Ekman-balanced) wind arises from the balance between pressure-gradient, friction and Coriolis forces, which is often a good approximation within the boundary layer (whereas geostrophic balance is not).

Explicit numerical resolution of boundary-layer flows in operational weather and climate models is impracticable, so specialized parametrizations are always used. There is then an issue about how to couple the parametrization to the dynamical core; this is normally done by forcing the right-hand side of the momentum equations, and the flow then has to react to this forcing. Forcings are also added to the thermodynamic equations. Compared with the research devoted to developing the dynamical core and parametrizations separately in weather and climate models, there is less research into how they are coupled (Staniforth *et al.*2002). In this article, we argue that the use of numerical solutions of the SGT equations (herein referred to as the SGT model) can expose potential deficiencies in existing methods of coupling the boundary layer and the dynamical core. The SGT model isolates the interaction of the boundary layer and the balanced dynamics. For example, Cullen (1989) compared the performance of a SGT model with a hydrostatic primitive (HP) model for a diurnally varying sea breeze. Whilst the SGT model followed the diurnal cycle of the heating with a time-lag due to friction, the HP model had difficulty in modelling the decay of the sea breeze. Given that the flow should be strongly tied to the diurnal heating variation, Cullen (1989) argued that the SGT model was more realistic in timing the decay of the sea breeze.

In the literature, most solutions of the SG equations including a boundary layer have solved the momentum balance (consisting of the inertial acceleration, pressure-gradient, friction and Coriolis forces) for either constant or vertical profiles of momentum diffusivity and idealized prescriptions of the pressure gradient (Wu and Blumen 1982; Tan and Wu 1994; Bannon 1998; Tan 2001; Tan and Wang 2002). Cullen (1989) also included prognostic potential temperature and boundary-layer heating and did not prescribe the pressure gradient. Since the article of Cullen (1989), there have been developments in both SGT theory and boundary-layer parametrization schemes that motivate the formulation of a revised model. Cullen (2006) stated a modification to his original equation set that had a negative-definite energy equation and also included the inertial acceleration in the *x* direction. Cullen (1989) used a boundary-layer parametrization that was a function of the local wind and temperature; since then, Holtslag and Boville (1993) have identified that local schemes were poor at developing the observed well-mixed thermodynamic profiles for convective boundary layers. Holtslag and Boville (1993) proposed a boundary-layer scheme that was both non-local (constrained by the depth of the convective boundary layer) and had counter-gradient terms (in order to maintain well-mixed thermodynamic profiles). The vertical resolution of Cullen (1989) also meant that only two or three grid points were in the boundary layer.

The aim of this article is to revise the SGT model, particularly to include well-mixed boundary layers. We formulate a numerical solution method for the more complete equation set of Cullen (2006). We adapt the predictor– corrector method used by Cullen (1989) for this equation set; a preconditioning stage calculating the geotriptic wind permits much higher vertical resolution than before; the elliptic corrector stage of Cullen (1989) is modified for well-mixed boundary layers; finally the boundary-layer parametrization of Holtslag and Boville (1993) is used. The revised model will give access to a wider range of scenarios in future applications and greater relevance to state-of-the-art weather and climate models. We investigate the benefits of the revised SGT model by comparing it with a HP model for a diurnally varying sea breeze.

### 2. The primitive and semi-geotriptic models

- Top of page
- Abstract
- 1. Introduction
- 2. The primitive and semi-geotriptic models
- 3. Parametrizations and numerical methods
- 4. Results
- 5. Discussion
- References

In this section, we review the approximations of the hydrostatic primitive equations that give the semi-geotriptic equations of Cullen (2006). For the purposes of simplifying the problem whilst still retaining advective dynamics and boundary-layer diffusivity, we reduce the equations to a two-dimensional form by neglecting *y* derivatives. Although a three-dimensional model should be seen as an ultimate goal, we decided to pursue an incremental development path from the two-dimensional model of Cullen (1989). The equations are solved in a domain defined by

- (1)

where *x* is the horizontal Cartesian coordinates and *z* the pseudo-height (Hoskins and Bretherton 1972). Note that Cullen (1989) used a sigma vertical coordinate.

#### 2.1. The hydrostatic primitive equations

The two-dimensional, Boussinesq, hydrostatic primitive equations on an *f*-plane, including boundary-layer terms, are

- (2)

- (3)

- (4)

- (5)

- (6)

- (7)

where D/(D*t*), *t*, (*u*, *v*, *w*), *f*, ϕ, *K*_{m}, *K*_{ h}, *F*_{θ}, *g*, θ, θ_{0} and *d*_{h} are the substantive derivative, time, wind vector, Coriolis parameter, geopotential, boundary-layer momentum diffusivity, boundary-layer heat diffusivity, boundary-layer heating rate, gravitational acceleration, potential temperature, reference potential temperature and horizontal diffusivity of vertical velocity respectively. The diurnal cycle scenarios investigated here have horizontal scales of the order of 50 km, so the *f*-plane assumption is sufficient.

Hydrostatic balance () is no longer valid in well-mixed boundary layers, as it assumes that time-scales should be larger than 1/*N* where *N* is the Brunt– Väisälä frequency (which is not possible when *N* = 0). It is therefore appropriate to include a horizontal diffusion term in the subgrid model to represent non-hydrostatic effects (Eq. (5)). The term will have a regularizing effect on the elliptic corrector equation used later. The inclusion of viscosity in this way is similar to the method used by Xu (1989) for moist symmetric instability, where a similar problem of zero potential vorticity occurs. The equations are solved with the boundary conditions

- (8)

- (9)

The boundary-layer depth (*h*) above which *K*_{m} is zero is such that *h* < *H*. The potential temperature (θ) is fixed at *z* = *H*, and its surface forcing will be defined later.

#### 2.2. Geotriptic balance

Scaling Eqs (2) and (3) for a typical horizontal velocity (*U*), horizontal length (*L*), vertical depth (Δ) and time-scale (τ) gives the Rossby (*R*_{o} =*U*/(*fL*)) and Ekman (*E*_{k} = *K*_{m}/(*f*Δ^{2})) numbers. For small Rossby numbers but significant Ekman numbers, and also assuming , Eqs (2) and (3) reduce to a balance between the Coriolis, geopotential gradient and boundary-layer forces:

- (10)

- (11)

where is the horizontal ‘geotriptic’ (Earth-rubbing) wind; is sometimes also called the Ekman-balanced wind (hence the subscript ‘e’). Taking the *z* derivative of Eq. (11) and substituting from the *x* derivative of Eq. (5) gives frictional thermal wind balance:

- (12)

where is the operator

When the boundary-layer momentum diffusivity is zero, Eqs (10) and (11) revert to geostrophic balance. When *d*_{h} = 0 and the boundary-layer momentum diffusivity is zero, Eq. (12) reverts to the usual thermal wind balance. When *f* = 0, Eqs (10) and (11) are still valid, but the left-hand side of Eq. (12) changes to a function of *u*_{e}. The boundary conditions on the geotriptic wind are

- (13)

#### 2.3. The semi-geotriptic momentum equations

When the time-scales are greater but not much greater than , and the Rossby number is significant, the substantive derivatives of momentum need to be included. In the semi-geotriptic equations, this is done via the ‘geotriptic momentum’ assumption, where the momentum is approximated by its geotriptic value in the substantive derivative:

- (14)

Motivated by forming an energy integral that decayed with time, Cullen (2006) proposed the following approximation of the momentum diffusivity term:

- (15)

Applying the above approximations to Eqs (2) and (3) gives the semi-geotriptic momentum equations:

- (16)

- (17)

Equations (4)– (7) are common to both the SGT and HP formulations. It is worth commenting on the difference between the SGT and HP horizontal momentum equations. Whilst (*u*, *v*) are prognostic variables in the HP horizontal momentum equations (Eqs (2) and (3)), the prognostic variables in the SGT horizontal momentum equations (Eqs (16) and (17)) are now the geotriptic wind (*u*_{e}, *v*_{e}). In the SGT equations, (*u*, *v*) are thus implicit, diagnostic quantities. It is this difference that gives a fundamentally different coupling between the boundary layer and the dynamics in the HP and SGT formulations. In the HP equations, (*u*, *v*) responds in a prognostic way to the Coriolis, pressure-gradient and boundary-layer drag forces; in the SGT equations, the inertial terms are constrained by geotriptic balance (Eq. (14)) and (*u*, *v*) adjusts to maintain the approximated momentum balance. Whilst the approximations in Eq. (15) are counterintuitive, they are justified by considering the domain-integrated energy (*E*_{int}) in the absence of heating:

- (18)

where

- (19)

The approximation in Eq. (15) thus forces the domain-integrated energy to be negative-definite (decaying with time, Eq. (18)), and is thus a more appropriate approximation of the boundary-layer drag than other options (such as for the *x*-component, used by Bannon (1998)). Since the equations are written relative to the rotating Earth, the frictional part of the boundary-layer model is representing the transfer of energy from the atmosphere to the solid Earth and should represent an energy sink. Compared with Cullen (2006), the right-hand side of Eq. (18) now contains additional negative-definite terms in vertical velocity (originating from Eq. (5)). Cullen (1989) showed that the diagnostic calculation of (*u*, *v*) in the SGT model can have benefits over the HP model for the diurnal cycles of sea breezes. Whilst the SGT model followed the diurnal cycle of the heating with a time-lag due to friction, the HP model had difficulty in modelling the decay of the sea breeze.

### 3. Parametrizations and numerical methods

- Top of page
- Abstract
- 1. Introduction
- 2. The primitive and semi-geotriptic models
- 3. Parametrizations and numerical methods
- 4. Results
- 5. Discussion
- References

In this section we define a boundary-layer parametrization that maintains well-mixed convective boundary layers. A new numerical solution method is then given, which accommodates the well-mixed layers.

#### 3.1. Boundary-layer parametrization

Holtslag and Boville (1993) demonstrated that diffusivity schemes that are a function of local variables are less effective at maintaining well-mixed convective boundary layers. Such a local scheme was used by Cullen (1989). Therefore, for our revised model, we use the formulation of Holtslag and Boville (1993) for heating by convective boundary layers. This scheme is chosen as it has performed well in the operational UK Met Office weather and climate models (Lock *et al.*2000). The convective boundary-layer heating is given by

- (20)

where γ is a counter-gradient term designed to maintain well-mixed potential temperature profiles. The boundary-layer diffusivity profiles are also non-local (controlled by the depth of the mixed layer) in convective boundary layers:

- (21)

where *Pr*, κ, *w*_{t} and *w*′θ′_{0} are the Prandtl number, Karman constant, a velocity scale dependent on shear and heat flux and surface heat flux respectively. γ and *w*_{t} are defined by Holtslag and Boville (1993). For stable and neutral boundary layers, γ = 0 and the diffusivity function of Brost and Wyngaard (1978) is used:

- (22)

where *u*_{*} and ϕ_{m} are the friction velocity and momentum stability function respectively. Beare *et al.* (2006b) showed that this parametrization performed well when compared with a range of stable boundary-layer large-eddy simulations. The surface momentum and heat diffusivities are calculated using the surface exchange scheme of Beljaars and Holtslag (1991).

#### 3.2. Numerical method for high-resolution well-mixed boundary layers

Cullen (1989) used a predictor– corrector method for the numerical solution of the SGT equations. The prognostic equations were advanced one time step (the predictor stage) and then *u* and *w* were corrected to bring the prognostic variables back into frictional thermal wind balance. The predictor and corrector stages were iterated to convergence at each time step. The corrector stage was a second-order partial differential equation with coefficients requiring strict static stability (∂θ/∂ *z* > 0) and strict frictional inertial stability (defined below in Eq. (28)) to be strictly elliptic and thus solvable numerically. The adjective ‘strict’ is used as, whilst some analytic solutions of the SG equations exist for the non-strictly elliptic case (Loeper 2006), general numerical solutions require strict ellipticity.

Enforcing strict static stability is a significant limitation, as the well-mixed boundary layer (∂θ/∂*z* = 0) is a very common feature in real applications. When using a parametrization of the form of Eq. (20), adjusting the vertical profile of θ to be statically stable at one time step (as in Cullen, 1989) will be counteracted by the boundary-layer scheme within a few time steps. Therefore, here a different approach is adopted where, instead, a horizontal diffusion of vertical velocity is included in Eq. (5) to form a strictly elliptic corrector equation for well-mixed boundary layers. In addition, the more complete SGT equations described previously are solved; also, a preconditioning stage is incorporated to accelerate convergence at higher vertical resolution.

The vertical velocity (*w*) was initialized to zero and the initial geopotential (ϕ) calculated from Eq. (5) using a prescribed initial potential temperature distribution (defined in section 3.3). Given θ at some time *t*, the sequence of a subsequent time step is as follows.

(1) Calculate the boundary-layer heating rate (*F*_{θ}) and momentum diffusivity (*K*_{m}) from Eqs (20)– (22).

(3) Preconditioning stage. Set *u* = *u*_{e} and calculate *w* from Eq. (6).

(4) Diagnose ∂ϕ/∂*x* by first vertically integrating Eq. (5) and then taking the *x*-derivative.

(5) Starting with values of *v*, *u* and θ from the previous time step, perform a single time step of the discretized prognostic equations, Eqs (16), (17) and (4), giving , and θ^{#}.

(6) Calculate increments (*u*^{′}, *w*^{′}) to correct (*u*, *w*) using the elliptic solver, Eq. (25), defined later in this subsection.

(7) Calculate *v* from the *x*-momentum balance, Eq. (24).

Steps (4)– (7) are an inner loop, which is iterated until (*u*, *v*, *w*) converges. Steps (1)– (3) are an outer loop, which is iterated (if required) until *K*_{m} and *F*_{θ} converge. The preconditioning (step (3)) was required to accelerate convergence at the higher vertical resolutions used here. Equations (11) and (17) are solved by first rearranging them in the following form:

- (23)

- (24)

Given that the momentum diffusivity is fixed over the inner loop, when Eqs (23) and (24) are discretized becomes a sparse banded matrix. The banded matrix was inverted using the lapack sparse matrix routines (Anderson *et al.*1999). The numerical solution of Eq. (23) for the idealized limit of constant *K*_{ m} and constant pressure-gradient was close to the analytic Ekman boundary-layer solution (Holton 1992).

The corrector equation is formed by linearizing Eqs (4), (5), (10), (11), (16) and (17) for increments (*u*^{′}, *w*^{′}), and retaining only terms in and , where ψ is the stream function.

- $$C_{zz}{\partial^{2}\psi \over \partial z^{2}} +\left(N^{2}-{d_{\rm h} \over \Delta t} {\partial^{2} \over \partial x^{2}}\right)\ {\partial^{2}\psi \over \partial x^{2}}={R \over \Delta t},$$ where (25)

- (26)

with boundary conditions

- (27)

*R* is the residual from frictional thermal wind balance, and the increments in Eq. (25) act to bring the prognostic variables back into thermal wind balance. *w*^{#} is the value of *w* from the previous iteration. The cross and single derivatives of ψ are excluded so that the increments are underestimated with each iteration of the inner loop. The values of *K*_{m}, *u*_{ e} and *v*_{e} in *C*_{zz} are evaluated from the previous time step. The stream-function formulation of the increments is made possible by continuity (Eq. (6)). Compared with Cullen (1989), *C*_{zz} now contains an additional term in ∂ *u*_{e}/∂*x*, as accelerations in the *x*-direction are now included. *C*_{zz} also includes a term that is the square of the vertical diffusivity operator, which originates from the approximation of the momentum diffusion term in Eq. (15). The condition for Eq. (25) to be strictly elliptic is

- (28)

where *E*_{ii} and *H*_{ii} are the diagonal components of the discretized operators

and

respectively. Strict frictional inertial stability means that the contents of the first bracket in Eq. (28) is positive. Equation (28) is less restrictive than the condition for the corrector used by Cullen (1989), as now boundary layers with *N* = 0 are permitted. Equation (25) is solved using an alternating-direction implicit (ADI) method (Press *et al.* 2007), using LAPACK sparse matrix inverters for the horizontal and vertical sweeps of the solver.

Apart from the corrector and preconditioning stages used for the SGT model, the HP and SGT models use identical discretizations and numerical schemes. The spatial discretization in each was Charney– Phillips in the vertical (horizontal momentum and potential temperature on different vertical half-levels, first potential temperature level at the surface) and Arakawa B in the horizontal (horizontal momentum and potential temperature staggered in the horizontal). The vertical velocity was held on the same points as potential temperature. The domain parameters were *L* = 200 km, *H* = 10 km. A horizontal grid length of 4 km was used, as in Cullen (1989). However, the vertical grid length was 100 m, giving approximately four times the number of levels within the boundary layer as was used by Cullen (1989). A time step of 6 s was used for both SGT and HP models. The advection terms in Eqs (2), (3), (16) and (17) were calculated using a semi-Lagrangian scheme with cubic-Lagrange interpolation, following Riishojgaard *et al.* (1998). The boundary-layer heating and momentum increments were calculated using the time-implicit Crank– Nicolson method (Press *et al.*2007). *d*_{ h} was set to 10^{4} m^{2}s^{−1}.

#### 3.3. Sea-breeze test case

In order to compare the SGT formulation with the HP model, we maintain continuity with the study of Cullen (1989) by comparing them for a diurnally varying idealized sea-breeze forcing. Once the new model is understood in this context, there is scope for other interesting cases (e.g. frontogenesis) in the future. The initial potential temperature distribution is uniformly stratified and uniform in *x* with a potential temperature lapse rate of 2×10^{−3} km^{−1}. The surface potential temperature (θ_{surf}) is given by

- (29)

where θ_{init} = 290 K, τ_{heat}= 24 h, *L*_{x} = 20 km, θ_{amp} = 5 K. This forcing gives a diurnal cycle over the land (*x* < 50 km), where the boundary layer is initially stably stratified, changing to convective by day and returning to stably stratified by night. For large positive values of *x* (*x* > 50 km), the sea location, θ_{surf}, tends to a constant value, θ_{init}. The resulting horizontal pressure gradient forces a diurnally varying sea breeze. This is an idealized forcing designed to expose the difference in the SGT and HP models. The horizontal tanh-function dependence is an addition to Cullen (1989), and was used to ensure that the horizontal scale of *v*_{e} and *u*_{e} obeyed frictional inertial stability (i.e. the contents of the first bracket in Eq. (28) were positive). The removal of the horizontal adjustment of *v*_{e} used by Cullen (1989) in the SGT model permitted a more precise comparison between HP and SGT models. The night-time cooling over land in this case is not sufficient to generate a night-time land breeze, which is sometimes observed (Stull 1988). Since the focus of this study is the growth and decay of the convective boundary layer and the associated sea breeze, a land breeze is not required (the SGT model can generate a land breeze when the night-time cooling is sufficient to reverse the horizontal pressure gradient). Both the HP and SGT models were run for 36 hours and *f* = 10^{−4} s^{−1}.

### 4. Results

- Top of page
- Abstract
- 1. Introduction
- 2. The primitive and semi-geotriptic models
- 3. Parametrizations and numerical methods
- 4. Results
- 5. Discussion
- References

Figure 1 compares time series of < *u*^{2}> and < *v*^{2}> for the SGT and HP models (where triangular brackets indicate a domain-average). In the first 11 hours, the < *u*^{2}> evolution of both models is close. However, < *v*^{2}> grows more rapidly for the SGT model than for the HP model. This is due to the balance constraint in the SGT model, with the Coriolis (*fv*) balancing the pressure-gradient, friction and scaled acceleration (as we will show later from the momentum budgets). Beyond 11 hours, there is significant divergence in the evolution of the SGT and HP models, particularly in < *u*^{2}>. The SGT model < *u*^{2}> peaks at 15 hours but the HP model value peaks much later at 19 hours, with 2.5 times the magnitude. Whilst the SGT model < *u*^{2}> decays following the transition between the convective and stable boundary layers (Figure 1(a) and (b)), the HP model decays slowly, in an oscillatory fashion, and with significant amplitude into the following day (24 hours). Observed sea breezes tend to decay following the surface forcing (Stull 1988) and not persist into the following day, so the slow decay of the HP model is physically unrealistic. Such unrealism may be in part due to the two-dimensional domain, with three-dimensional simulations possibly providing additional drag; nevertheless the benefits of the balanced assumption in the SGT model are very apparent. The < *v*^{2}> values of the SGT model beyond 16 hours decay slightly, but the HP values continue to grow.

Figure 2 shows the time evolution of vertical cross-sections of potential temperature for the SGT and HP models. The potential temperature distributions play the main role in determining the pressure gradient distribution (Eq. (5)). At 12 hours, both model formulations show a well-mixed boundary layer of depth 1 km over land (*x* < 50 km), sustained by non-local boundary-layer mixing (Eqs (20) and (21)). Whilst, in Cullen (1989), further adjustment of potential temperature was required to maintain strict static stability, here the horizontal diffusion term (Eq. (5)) permits the potential temperature to be well mixed. The potential temperature distributions in both HP and SGT models are close at 12 hours, giving similar pressure gradient forces (see momentum budgets later). The sea-breeze front is between *x* = − 25 km and *x* = 25 km at 12 hours. By 24 hours (Figure 2(b) and (e)), the boundary layer is stable (Figure 1(a)) and the sea-breeze front moves further inland. The inland progression and strength of the sea-breeze front at 24 hours is more significant in the HP model as the sea-breeze circulation is stronger. By 36 hours (Figures 2(c) and (f)), both HP and SGT models have re-established a well-mixed boundary layer over land, but with the sea-breeze front about 15 km to the west of its location at 12 hours.

The diurnal cycle of the sea-breeze circulation is shown in Figure 3. At 12 hours, the SGT model show a sea-breeze circulation in the lowest 500 m with a peak of − 1.4 m s^{−1} (Figure 3(a)). There is a weaker positive return flow above this. The HP model circulation is similar (Figure 3(d)) in structure, with a slightly weaker sea breeze and stronger return flow. Such a close match in the early stages of a sea breeze was not found by Cullen (1989), as additional adjustment of the potential temperature was required in his SGT model.

Figure 4 shows that the vertical grid length of 100 m (compared with approximately 400 m used by Cullen (1989)) permits a better definition of the upper-level front and thus a a more tilted sea-breeze circulation at 12 hours. Moreover, Figure 5 shows that the non-local boundary-layer scheme (Eq. (20)) generates a more tilted circulation and weaker return flow compared with the local scheme of Cullen (1989). The tilted circulation is associated with a more well-defined upper-level front generated by the stronger vertical thermal mixing of the non-local scheme.

By 24 hours (Figure 3(b) and (e)), the sea-breeze circulations for both HP and SGT models have diverged significantly. The SGT model sea-breeze circulation is weak and very shallow, whereas the HP model sea breeze continues to be significant and to propagate inland. The SGT model clearly follows the boundary-layer state much more closely than the HP model, and this generates a more physically realistic evolution. By 36 hours (Figure 3(c) and (f)), the sea-breeze circulations of the HP and SGT models are similar between *x* = − 50 km and *x* = 50 km, but the sea-breeze circulation extends much further over land in the HP model. Whilst the SGT model evolution is much more periodic in time, the HP model generates an unphysical residual circulation that persists into the following day.

The *v* cross-sections for the HP and SGT models are shown in Figure 6. In the SGT simulation, the distribution of *v* was close to *v*_{ e}, so the distribution of *v* in the SGT model can be understood predominantly with reference to the frictional thermal wind balance (Eq. (12)). At 12 hours, the horizontal potential temperature gradients (Figure 2(a)) are positive over land (*x*<50 km) and below the top of the well-mixed boundary layer, but negative near the top of the well-mixed boundary layer. In addition, the boundary-layer momentum diffusivity, and thus , increases with the thermal boundary-layer diffusivity between *x*=50 km and *x*=−50 km. Equation (12) shows that *v*_{e} follows these potential temperature gradients, being positive in the lowest 1 km and negative above, with a symmetry determined by the inversion of the well-mixed boundary layer in Figure 2(a). The *v*_{e} distribution is also influenced by the strong momentum diffusivity (*K*_{m} and thus ), becoming progressively well-mixed over land (Figure 6(a)). This effect was much weaker in Cullen (1989) due to the use of a local boundary-layer diffusivity formulation. At 48 hours the boundary-layer diffusivity is much weaker in the SGT model (due to stable stratification) so and the *v* distribution follows the distribution of potential temperature (Figure 2(b)) more closely. By 36 hours, the *v* distribution recovers a similar structure to that at 12 hours, except that the horizontal extent of the near-surface *v* extends over the sea slightly. This is due to the slight descent of the isentropes for 0 km < *x*< 100 km (Figure 2(c)).

In the HP model, the *v* distribution is not constrained by frictional thermal wind balance, giving contrasting results. At 12 hours (Figure 6(d)), the *v* field is weaker than the SGT model as it needs to be formed by the action of the Coriolis force, *fu* (Eq. (3)), rather than a balance constraint. By 24 hours, the *v* field is stronger than the SGT model near the surface, and has advanced more inland, following the sea-breeze circulation (Figure 3(e)). By 36 hours, *v* has a greater horizontal extent than in the SGT model (Figure 6(f)); the 36 hour *v* of the HP model is also not as close to its 12 hour distribution as in the SGT model.

In order to clarify the similarities and differences between the SGT and HP dynamics, Figure 7 compares the momentum budgets in the *x*-direction over a land point for 12 hours (when the surface temperature is maximum) and 18 hours (when the HP sea-breeze circulation is maximum). The budgets at 12 hours (Figure 7(a) and (c)) are similar: the pressure gradient and friction are the most significant terms and approximately balance. The *x*-acceleration and Coriolis terms are a small component in the balance. The *y*-acceleration was also small relative to other terms in the *y*-momentum balance. For this reason, the corrector increment (*u*^{′}, not shown), which corrects for accelerations, was small relative to *u*_{e}. By 18 hours, the differences in the SGT and HP models are revealed by the momentum balance. In the SGT model, an approximate balance between Coriolis and pressure gradient is present above the stable boundary layer (200 m), but in the HP model the balance is between the tendency and Coriolis terms. The residual sea-breeze circulation in the HP model at 18 hours is thus due to inertial oscillations.

### 5. Discussion

- Top of page
- Abstract
- 1. Introduction
- 2. The primitive and semi-geotriptic models
- 3. Parametrizations and numerical methods
- 4. Results
- 5. Discussion
- References

In this article we described a modification of the semi-geostrophic model to include well-mixed boundary layers. Both the substantive derivative and the momentum diffusion term were approximated using the geotriptic (Ekman-balanced) wind to ensure a negative-definite energy evolution in the absence of heating (Cullen 2006). Unlike Cullen (1989), the acceleration in the *x*-direction was included for completeness. The well-mixed thermodynamic profiles were included by a modification of the hydrostatic balance to include a horizontal diffusion of vertical velocity. Such an approach is similar to that used by Xu (1989) for moist symmetric instability. Given the ubiquity of well-mixed boundary layers in real flows, the new SGT model will have a larger range of applicability.

The numerical solution of the SGT equations involved a modification of the predictor– corrector method of Cullen (1989). The corrector stage now includes the horizontal diffusion term of vertical velocity to ensure strict ellipticity. Also, the vertical coefficients of the elliptic corrector equation were modified to incorporate the approximation of the boundary-layer momentum diffusion term introduced by Cullen (2006). Convergence at higher vertical resolution was accelerated by a preconditioner calculating the Ekman-balanced wind. The well-mixed boundary layers were produced using the non-local boundary-layer diffusivity scheme of Holtslag and Boville (1993).

The two-dimensional SGT and HP models were compared for an idealized sea-breeze diurnal cycle. The inclusion of the well-mixed boundary layers meant that the sea-breeze circulations were much closer in the initial growth phase (12 hours) than Cullen (1989). This was because the adjustment of potential temperature to maintain strict static stability in the SGT model was not required and thus the pressure gradient forces were similar between the models. The decay phase was in agreement with Cullen (1989) in that the semi-geotriptic (SGT) model followed closely the boundary-layer transition between convective and stable states; the hydrostatic primitive (HP) model had an unrealistic additional circulation during the night-time stable phase. The vertical resolution was approximately four times that used by Cullen (1989), which was shown (by comparison with coarser vertical resolution) to better resolve the inversion at the top of the mixed layer, with a tilting of the sea-breeze circulation along it.

Momentum budgets revealed that the additional circulation in the HP model at night was due to inertial oscillations. Whilst there are sometimes inertial oscillations in very high-resolution simulations of evening-transition boundary layers (Beare *et al.*2006a), in the HP model their magnitude and persistence into the following day were unrealistic. There is no evidence in the literature that the sea breeze should persist through the night (Stull 1988). The two-dimensional domain of the HP model may account for some of the lack of decay, with three-dimensional decay effects generating additional dissipation. However, another candidate for the differences between the SGT and HP models is the resolution used. The HP model will be, by definition, more accurate than the SGT model at very high resolutions, when the boundary layer is no longer parametrized. However, at the resolutions used here (typical of numerical weather prediction), the HP model is coupled to a parametrized boundary layer. The simpler structure of the SGT model, with explicit enforcement of balance, demonstrates a better coupling to the boundary-layer parametrization at the resolution used.

The momentum budgets also showed that, for the horizontal scales and heating used in the sea breeze (similar to Cullen (1989)), the *x*- and *y*-accelerations were small components relative to boundary-layer drag and friction. This is a feature of the chosen sea-breeze scenario (picked to maintain continuity with Cullen (1989)) being controlled by the boundary layer, which generates both the pressure gradient (from the heating) and the boundary-layer drag. The SGT model would thus benefit from being tested in future in other scenarios of mixed layers with larger momentum accelerations (e.g. forcing from geostrophic winds above the boundary layer).

Formulations of numerical weather prediction and climate models divide conveniently into resolved dynamical-core and subgrid physical parametrizations. Improving these components is an ongoing area of research and development. Arguably less examined, however, is the coupling between these components. More research on the physics– dynamics coupling might therefore lead to significant improvements in future model performance. There are several approaches to understanding how the boundary-layer scheme couples to the dynamical core. One is to vary separately the controlling parameters of the boundary-layer scheme and the dynamical core for the full NWP model (Beare 2007). Whilst this shows the relative importance of aspects of the model components, the changes are still within a single dynamical framework; in order to understand the interactions further, there is a need to ‘pull apart’ the dynamical framework. Here, we demonstrated a new approach to physics– dynamics coupling. We compared a full dynamics model with a model explicitly constrained to balance the large-scale dynamics and the boundary-layer parametrization (the SGT model). The SGT model then provided a critical lense through which to view the full model. Here, the balanced SGT model exposed issues with the ability of the HP model to decay circulations forced by a diurnal cycle.

### References

- Top of page
- Abstract
- 1. Introduction
- 2. The primitive and semi-geotriptic models
- 3. Parametrizations and numerical methods
- 4. Results
- 5. Discussion
- References

- 1999. lapack Users' Guide, 3rd edn. SIAM; 407pp. , , , , , , , , , , .
- 1998. A comparison of Ekman pumping in approximate models of the accelerating planetary boundary layer. J. Atmos. Sci. 55: 1446–1451. .
- 2007. Boundary layer mechanisms in extratropical cyclones. Q. J. R. Meteorol. Soc. 133: 503–515. .
- 2006a. Simulation of the observed evening transition and nocturnal boundary layers: Large-eddy simulation. Q. J. R. Meteorol. Soc. 132: 81–99. , , .
- 2006b. An intercomparison of large-eddy simulations of the stable boundary layer. Boundary-Layer Meteorol. 118: 247–272. , , , , , , , , , , , , , , , , .
- 1991. Flux parameterization over land surfaces for atmospheric models. J. Appl. Meteorol. 30: 327–341. , .
- 1978. A model study of the stably stratified planetary boundary layer. J. Atmos. Sci. 35: 1427–1440. , .
- 1989. On the incorporation of atmospheric boundary layer effects into a balanced model. Q. J. R. Meteorol. Soc. 115: 1109–1131. .
- 2006. A Mathematical Theory of Large-Scale Atmosphere/Ocean Flow. Imperial College Press: London; 259pp. .
- 1992. An Introduction to Dynamic Meteorology. Academic Press: New York; 507pp. .
- 1993. Local versus nonlocal boundary-layer diffusion in a global climate model. J. Climate 6: 1825–1842. , .
- 1975. The geostrophic momentum approximation and semi-geostrophic equations. J. Atmos. Sci. 33: 233–242. .
- 1972. Atmospheric frontogenesis models: mathematical formulation and solution. J. Atmos. Sci. 29: 11–37. , .
- 2000. A new boundary layer mixing scheme. Part I: Scheme description and single-column model tests. Mon. Weather Rev. 128: 3187–3199. , , , , .
- 2006. A fully nonlinear version of the incompressible Euler equations: the semi-geostrophic equations. SIAM J. Math. Anal. 38: 795–823. .
- 2007. Numerical Recipes: the Art of Scientific Computing, 3rd edn. Cambridge University Press: Cambridge, UK; 1256pp. , , , .
- 1998. The use of spline interpolation in semi-Lagrangian transport models. Mon. Weather Rev. 126: 2008–2016. , , , .
- 2002. Analysis of the numerics of physics–dynamics coupling. Q. J. R. Meteorol. Soc. 128: 2779–2799. , , .
- 1988. An Introduction to Boundary Layer Meteorology. Kluwer Academic: Dordrecht; 666pp. .
- 2001. An approximate analytical solution for the baroclinic and variable eddy diffusivity semi-geostrophic Ekman boundary layer. Boundary-Layer Meteorol. 98: 361–385. .
- 2002. Wind structure in an intermediate boundary layer model based on Ekman momentum approximation. Adv. Atmos. Sci. 19: 266–278. , .
- 1994. The Ekman momentum approximation and its application. Boundary-Layer Meteorol. 68: 193–199. , .
- 1982. An analysis of Ekman boundary layer dynamics incorporating the geostrophic momentum approximation. J. Atmos. Sci. 39: 1774–1782. , .
- 1989. Extended Sawyer–Eliassen equation for frontal circulations in the presence of small viscous moist symmetric instability. J. Atmos. Sci. 46: 2671–2683. .