Use of a Mixed-Layer Model to Investigate Problems in Operational Prediction of Return Flow

John M. Lewis National Severe Storms Laboratory, Norman, Oklahoma, and Desert Research Institute, Reno, Nevada

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Abstract

Inaccuracy in the numerical prediction of the moisture content of return-flow air over the Gulf of Mexico continues to plague operational forecasters. At the Environmental Modeling Center/National Centers for Environmental Prediction in the United States, the prediction errors have exhibited bias—typically too dry in the early 1990s and too moist from the mid-1990s to present. This research explores the possible sources of bias by using a Lagrangian formulation of the classic mixed-layer model. Justification for use of this low-order model rests on careful examination of the upper-air thermodynamic structure in a well-observed event during the Gulf of Mexico Experiment. The mixed-layer constraints are shown to be appropriate for the first phase of return flow, namely, the northerly-flow or outflow phase. The theme of the research is estimation of sensitivity—change in the model output (at termination of outflow) in response to inaccuracies or uncertainties in the elements of the control vector (the initial conditions, the boundary conditions, and the physical and empirical parameters). The first stage of research explores this sensitivity through a known analytic solution to a reduced form of the mixed-layer equations. Numerically calculated sensitivity (via Runge–Kutta integration of the equations) is compared to the exact values and found to be most credible. Further, because the first- and second-order terms in the solution about the base state can be found exactly for the analytic case, the degree of nonlinearity in the dynamical system can be determined. It is found that the system is “weakly nonlinear”; that is, solutions that result from perturbations to the control vector are well approximated by the first-order terms in the Taylor series expansion. This bodes well for the sensitivity analysis. The second stage of research examines sensitivity for the general case that includes moisture and imposed subsidence. Results indicate that uncertainties in the initial conditions are significant, yet they are secondary to uncertainties in the boundary conditions and physical/empirical parameters. The sea surface temperatures and associated parameters, the saturation mixing ratio at the sea surface and the turbulent transfer coefficient, exert the most influence on the moisture forecast. Uncertainty in the surface wind speed is also shown to be a major source of systematic error in the forecast. By assuming errors in the elements of the control vector that reflect observational error and uncertainties in the parameters, the bias error in the moisture forecast is estimated. These bias errors are significantly greater than random errors as explored through Monte Carlo experiments. Bias errors of 1–2 g kg−1 in the moisture forecast are possible through a variety of systematic errors in the control vector. The sensitivity analysis also makes it clear that judiciously chosen incorrect specifications of the elements can offset each other and lead to a good moisture forecast. The paper ends with a discussion of research approaches that hold promise for improved operational forecasts of moisture in return-flow events.

Corresponding author address: John Lewis, National Severe Storms Laboratory, 120 David L. Boren Blvd., Norman, OK 73072. Email: jlewis@dri.edu

Abstract

Inaccuracy in the numerical prediction of the moisture content of return-flow air over the Gulf of Mexico continues to plague operational forecasters. At the Environmental Modeling Center/National Centers for Environmental Prediction in the United States, the prediction errors have exhibited bias—typically too dry in the early 1990s and too moist from the mid-1990s to present. This research explores the possible sources of bias by using a Lagrangian formulation of the classic mixed-layer model. Justification for use of this low-order model rests on careful examination of the upper-air thermodynamic structure in a well-observed event during the Gulf of Mexico Experiment. The mixed-layer constraints are shown to be appropriate for the first phase of return flow, namely, the northerly-flow or outflow phase. The theme of the research is estimation of sensitivity—change in the model output (at termination of outflow) in response to inaccuracies or uncertainties in the elements of the control vector (the initial conditions, the boundary conditions, and the physical and empirical parameters). The first stage of research explores this sensitivity through a known analytic solution to a reduced form of the mixed-layer equations. Numerically calculated sensitivity (via Runge–Kutta integration of the equations) is compared to the exact values and found to be most credible. Further, because the first- and second-order terms in the solution about the base state can be found exactly for the analytic case, the degree of nonlinearity in the dynamical system can be determined. It is found that the system is “weakly nonlinear”; that is, solutions that result from perturbations to the control vector are well approximated by the first-order terms in the Taylor series expansion. This bodes well for the sensitivity analysis. The second stage of research examines sensitivity for the general case that includes moisture and imposed subsidence. Results indicate that uncertainties in the initial conditions are significant, yet they are secondary to uncertainties in the boundary conditions and physical/empirical parameters. The sea surface temperatures and associated parameters, the saturation mixing ratio at the sea surface and the turbulent transfer coefficient, exert the most influence on the moisture forecast. Uncertainty in the surface wind speed is also shown to be a major source of systematic error in the forecast. By assuming errors in the elements of the control vector that reflect observational error and uncertainties in the parameters, the bias error in the moisture forecast is estimated. These bias errors are significantly greater than random errors as explored through Monte Carlo experiments. Bias errors of 1–2 g kg−1 in the moisture forecast are possible through a variety of systematic errors in the control vector. The sensitivity analysis also makes it clear that judiciously chosen incorrect specifications of the elements can offset each other and lead to a good moisture forecast. The paper ends with a discussion of research approaches that hold promise for improved operational forecasts of moisture in return-flow events.

Corresponding author address: John Lewis, National Severe Storms Laboratory, 120 David L. Boren Blvd., Norman, OK 73072. Email: jlewis@dri.edu

1. Introduction

During the cool season (November–March), there is a rhythmic cycle of cold air penetrations into the Gulf of Mexico, on the order of four–five cycles per month (Crisp and Lewis 1992). These penetrations are associated with the migratory cyclone–anticyclone couplets that move across the United States. As the cyclone centers typically track eastward or northeastward from the Great Plains to the eastern seaboard, the associated anticyclonic centers track southeastward. On the backside (west side) of the cyclone and ahead (east) of the anticyclone, cold continental air streams toward the Gulf. To those along the Gulf coast, these strong north winds are called “northers.” As the anticyclone moves eastward and typically into the western Atlantic Ocean or the Caribbean, the southward streaming air over the Gulf sequentially turns westward and then northward—a clockwise turning and return to the land from whence it came and thus the label “return flow” (Henry 1979a, b).

On its outward-bound journey into the Gulf, continental air is warmed and moistened. The degree of airmass modification is dependent on the source region of the continental air—in essence, a dependence on the air–sea temperature difference. Independent of the source region, most of the modification takes place during the northerly flow or outflow phase. On its return to land, the marine-layer air experiences minimal warming and moistening (Lewis and Crisp 1992).

The returning air mass is typically entrained into the circulation of the next cyclone tracking across the United States. The numerical forecasts of intensity and movement of the synoptic systems associated with return flow have been remarkably good (Weiss 1992). In turn, the trajectories of low-level air have been faithfully represented, even over the data-sparse Gulf. Nevertheless, the thermodynamic characteristics of the returning air mass—its moisture content and the depth of the moist layer—have proved more problematical (Janish and Lyons 1992; Manikin et al. 2002). In the presence of erroneous moisture forecasts, there is uncertainty in the weather associated with return flow—a spectrum that includes fog/stratus, shallow convection, and severe storms.

The following statement extracted from an Environmental Modeling Center/National Centers for Environmental Prediction (EMC/NCEP)1 memorandum (Manikin 1998) captures the problems faced by meteorologists:

It has been noted during the past 3 cool seasons that the Eta model overdoes the return of moisture from the Gulf of Mexico and the Atlantic Oceans. . . . Moisture return is a key issue in convective forecasting, as subtle low-level moisture changes can be the difference between a stable sounding or one able to support deep convection. When the Eta was first instituted [in 1993],2 it was somewhat too slow in returning moisture (although superior to the NGM [Nested Grid Model]). But for several years now, forecasters like Steve Weiss of the Storm Prediction Center have noted that the Eta is now too fast. The problem is most pronounced at the leading edge of the deeper moisture.

In this memo, Manikin exhibits a set of low-level thermodynamic profiles from the Eta Model and concludes that “. . . turning off the shallow convection (Fig. 4) [Manikin’s Fig. 4, not shown] gives a better looking boundary layer than Fig. 1 [Manikin’s Fig. 1, not shown], so we can conclude that the parameterization of shallow convection is a major factor [in the moisture bias]” (Manikin 1998, p. 1). The shallow convection is a component of the Betts–Miller–Janjic (BMJ) convection scheme and it accounts for the upward and downward moisture transport in the presence of nonprecipitating shallow convective cloud (Janjic 1994, 2001). This shallow convection scheme cannot be the sole cause of the bias, however, since it was held constant between 1993 and 1998 (Z. Janjic 2006, personal communication) while the transition from dry to moist bias occurred in fall 1995 (Manikin 1998, p. 1).

An example of the moisture bias is seen by examination of NCEP forecasts from April 1999. In the top two panels of Fig. 1, we have (a) the 0000 UTC 19 April analysis of surface moisture and pressure along with wind observations from several buoys, and (b) the 24-h prediction of the surface moisture from NCEP’s Eta Model, verifying at 0000 UTC 19 April. Several forecasted wind vectors in the western Gulf are included in Fig. 1b. In the bottom two panels of Fig. 1, we have (c) the analysis at 0000 UTC 20 April, and (d) the 48-h forecast that verifies at 0000 UTC 20 April. The 24- and 48-h wind forecasts are reasonably good—capturing the anticyclonic structure of the wind field over the Gulf including the narrow band of strong southerly flow along the Gulf’s western boundary. However, the forecasted moisture along this western boundary exhibits positive bias of 1–2 g kg−1. As mentioned earlier and as confirmed by studies from the Gulf of Mexico Experiment (GUFMEX; Lewis et al. 1989), the bulk of the marine-layer moistening in the return flow occurs during the northerly-flow phase. Thus, the bias portrayed in the western Gulf is most likely the result of incorrect air–sea interaction processes that occurred at an earlier stage of the event.

It is challenging to modify the components of an operational prediction model in an effort to improve a particular output such as return-flow moisture. In essence, the nonlinear coupling between variables in the governing equations makes it difficult to conceptually determine the sensitivity of model output to the elements of the control vector (initial conditions, physical and empirical parameters, and boundary conditions). This is the justification for exploration of sensitivity based on powerful tools such as adjoint modeling (see, e.g., Errico 1997; Lewis et al. 2001). Even in the presence of tools such as the adjoint model, results often prove difficult to interpret. Research results from the Lewis et al. (2001) study indicated that air–sea interaction processes during the outflow phase were crucial to quality vapor flux forecasts along the Texas coast, yet the complex structure of the sensitivity fields proved puzzling. Specifically, areas of positive sensitivity (flux increases with incremental increases in SSTs along low-level trajectories) were juxtaposed with areas of negative sensitivity (flux increases with incremental decreases in SSTs along parallel low-level trajectories).

In the same spirit as found in the Lewis et al. (2001) study, we attempt to examine the sensitivity of moisture forecast to elements of the control vector. In contrast to the Lewis et al. (2001) study, however, we limit our study to the outflow phase of return flow. During this phase, the thermodynamics are well approximated by the low-order mixed-layer model; and instead of exploring sensitivity with a model of large dimensionality (the order of ∼105 elements), we use a model whose control vector has 30 elements. With this limited number of elements, it is unnecessary to use the sophisticated machinery of adjoint modeling.

Justification for use of the mixed-layer model rests on examination of the upper-air observations during a return-flow event on 21–23 February 1988—a well-observed case that occurred during project GUFMEX. Two Lagrangian boundary layer models have been successful in simulating this event—Burk and Thompson (1992) and Liu et al. (1992). Research results from these simulations gave creditable predictions over limited times and limited areas—a 12-h prediction over a swath that straddled the shelf in the Burk and Thompson (1992) study and a 5-h prediction over the Loop Current in the Liu et al. (1992) study.

We begin this research by analyzing observations of the thermodynamic structure in a Lagrangian frame of reference. This analysis is followed by a brief description of the mixed-layer model. By recourse to a “special case” analytic solution to this model, errors in estimates of the sensitivity are determined when the fourth-order Runge–Kutta (R–K) integration is used. The “general case” is then integrated via the R–K methodology and validated against the observations. Under the assumption of known systematic uncertainties in the elements of the control vector, estimates of the biased forecast errors are determined. The presentation concludes with a view of the biased operational forecasts in light of the sensitivity results from this study.

2. Case study

a. Upper-air observations and trajectory

During the 21–23 February 1988 event, a series of upper-air observations were made that capture the airmass transformation process. The rawinsondes were launched from Salvia, a U.S. Coast Guard ship whose duties in the Gulf included law enforcement, tending buoys, and search/rescue missions. She was able to keep pace with low-level outflow winds in both speed (∼10 m s−1) and direction (∼360°). Accordingly, these soundings represent a Lagrangian-framed view of the airmass modification process.

The first sounding from Salvia was taken at approximately 1200 UTC 21 February at a location just east of the Mississippi Delta. In Fig. 2, the small-blackened circle near the delta shows the location of this sounding. As Salvia traveled southward, soundings were taken at roughly 6-h intervals. Again, these locations are shown in Fig. 2 by the blackened circles. At the approximate midpoint along the line connecting the last two sounding locations of Salvia, a dropsonde was launched from a U.S. Air Force (USAF) reconnaissance aircraft. The location of this dropsonde is indicated by the blackened square in Fig. 2. For future reference, we chronologically number these soundings. The numerical ordering and time of the soundings are as follows: 1) 1138 UTC/21 (21 February), 2) 1728 UTC/21, 3) 0005 UTC/22, 4) 0300 UTC/22 (dropsonde), and 5) 0530 UTC/22.

The trajectory of surface air that most closely followed the path of Salvia is shown in Fig. 2. This trajectory is found by amalgamating the rich set of surface observations from commercial vessels (responsive to the special request for observations during GUFMEX), the National Data Buoy Center (NDBC) moored buoys, and observations from Salvia. Along this trajectory, the translation speed gradually decreases—from 12 m s−1 along the first third of the trajectory to 10 m s−1 along the middle third to 8 m s−1 along the final third of the trajectory. The trajectory was found by kinematic analyses (isotach and isogon analysis) at 6-h intervals following the procedures found in Saucier (1955, section 10.06). Observed surface winds (and pressure analyses) over the Gulf are shown in Fig. 3.

b. Sea surface temperatures

The broad view of the SST pattern for our case study is shown in the lower panel of Fig. 4. The upper panel depicts the bathymetry of the Gulf. Comparison of these two panels indicates that the colder water resides over the shelf and the warmest water is associated with the Loop Current in the eastern Gulf. This SST field is based on statistical regression—a pairing of the infrared radiance measurements against SSTs from instrumented buoys (Strong and McLain 1984; McLain et al. 1985).

The SST pattern shown in Fig. 4 is a composite of radiance data collected over a one-week period. Cloud contamination of these measurements dictates the use of a composite—a process that incorporates aged data to insure that SSTs stem from cloud-free retrievals. During the cool season, these radiance-derived SSTs exhibit errors as large as 2°C (Fredrickson and Lewis 1994).

In an effort to improve the radiance-derived SST pattern over the Gulf, the National Hurricane Center (NHC) developed a scheme that used this large-scale SST field as background and added detail by incorporating observations from buoys and ships of opportunity. This scheme was man–machine interactive using the Man-computer Interactive Data Access System (Suomi et al. 1983). Two or three times each week during the late 1980s–early 1990s, these computer-generated SST patterns were produced. In our study, we use the man–machine-generated SST field produced for 21 February 1988. Contours from this product are superimposed over the low-level trajectory shown in Fig. 2. Compared with the radiance-derived SST pattern, there is small-scale detail along the Gulf’s northern boundary that stems from NDBC buoy observations. A mesh of roughly 30-km spacing can resolve the detail in this analysis.

c. Observed thermodynamic structure

The Lagrangian evolution of the potential temperature θ and water vapor q are shown in Figs. 5 and 6, respectively. The profiles are numbered chronologically as discussed in section 2a. Profile 4 comes from the USAF aircraft, and because the humidity sensor failed during this dropsonde launch, the water vapor data are missing at this time.

The salient features of the potential temperature profiles are as follows:

  1. a superadiabatic layer, /dz < 0, exists at all times in the lowest 50–100 m,

  2. a layer of nearly constant θ exists above the superadiabatic layer and this temperature increases with time,

  3. the height of the constant-θ layer increases with time,

  4. immediately above the constant-θ layer, there is an abrupt positive jump in θ, and this jump monotonically increases with time, and

  5. above the jump, there is a layer of stable air where /dz < 0.

The salient features of the water vapor profiles are as follows:

  1. in the lowest 50–100 m, there is a rapid decrease of q with height,

  2. above this layer of rapid decrease, there is a layer of nearly constant q with evidence of a weak negative gradient of q near the top of layer,

  3. the height of the constant-q layer increases with time in the presence of layer moistening,

  4. immediately above the constant-q layer, there is an abrupt negative jump in q, and

  5. above the jump, there is a lapse of water vapor content with height.

Examination of the θ and q profiles indicates that the height of the constant-θ and constant-q layers are approximately equal at each time. The relative humidity (RH) increases from the surface to the top of the mixed layer for the thermodynamic profiles, yet saturation is never achieved. The greatest RH is 80%, observed at the top of the mixed layer at 0530 UTC 22 February. This is consistent with the absence of cloud along the trajectory (see Fig. 7).

In the earlier study by Lewis and Crisp (1992), subsidence above the mixed layer in this case was found by integrating the mass conservation equation. Twelve upper-air stations (three over the Gulf and nine along the U.S. and Mexican coastlines) were used to make the calculation, and results indicated that the subsidence over the low-level trajectory shown in Fig. 2 was 1–2 cm s−1 [∼1 cm s−1 at 850 mb (1.5 km) and ∼2 cm s−1 at 700 mb (3.0 km)].

d. Observed wind structure

The vertical profiles of wind, obtained from the rawinsonde launches aboard Salvia, are shown in Fig. 8. In contrast to the thermodynamic profiles, the wind profiles do not exhibit classic mixed-layer structure where a uniform distribution is bounded by discontinuities above and below the layer (Tennekes and Driedonks 1981). The wind profiles at the last two times exhibit more uniformity than at the earlier times, but there is no discontinuity at the top of the mixed layer (where we assume the top is at ∼1.5 km). The surface anticyclone is centered over Salvia at 0530 UTC February 22, and this gives rise to weak winds throughout the lower troposphere at the ship’s location.

3. Mixed-layer model: Lagrangian formulation

During the outflow phase of return flow, the cold, dry air of continental origin is heated from below, and turbulent transfer of heat, momentum, and water vapor as well as the other chemical species takes place. The process is referred to as “penetrative convection”—the advance of a turbulent fluid into a layer of stable stratification (Deardorff et al. 1969). The turbulence erodes the stable layer and ideally produces a layer typified by constant potential temperature, a uniform/constant wind, and constant mixing ratios of the chemical constituents including water vapor. Accordingly, this layer is labeled the “mixed layer.” In the case of return flow, the buoyancy that builds the mixed layer upward is counteracted by large-scale subsidence. The wind is viewed as the mechanism that translates the thermal plumes and associated turbulence downstream (Willis and Deardorff 1975)—the translation velocity for the Lagrangian frame in our case. A pedagogical review of mixed-layer modeling and research is found in Tennekes and Driedonks (1981).

Figure 9 schematically depicts the thermodynamic structures associated with the classic mixed-layer model or 1D slab model as it is often called (Tennekes and Driedonks 1981). Turbulence is assumed sufficient to maintain vertically uniform distributions of the variables in the presence of discontinuities at the upper and lower boundaries of the layer. The distributions shown in Figs. 5 and 6 closely approximate the ideal structure. Terms in the governing equations are defined as follows:

  • θ: Potential temperature of the mixed layer

  • H: Height of the mixed layer

  • q: Water vapor concentration in mixed layer

  • σ: Jump in potential temperature at the top of the mixed layer

  • μ: Jump in water vapor content at the top of the mixed layer

  • θs: Potential temperature of air at the sea surface

  • γθ (=/dz): Stability (vertical gradient of temperature) above the mixed layer

  • γq (=dq/dz): Vertical gradient of water vapor above the mixed layer

  • w: Large-scale vertical velocity (positive upward)

  • we: Entrainment velocity at top of mixed layer (positive upward)

  • Vs: Speed of air immediately above air–sea interface

  • cT: Transfer coefficient (nondimensional)

  • κ: Entrainment coefficient (nondimensional)

The governing equations as found in Lilly (1987) and Liu et al. (1992) follow, where subscript s, that is, ( )s, denotes evaluation at the air–sea interface:
i1520-0493-135-7-2610-e1
i1520-0493-135-7-2610-e2
i1520-0493-135-7-2610-e3
i1520-0493-135-7-2610-e4
i1520-0493-135-7-2610-e5
where the entrainment velocity we is given by
i1520-0493-135-7-2610-e6
These equations are not exempt from closure problems that plague turbulent boundary layers. To obtain solutions, we parameterize the fluxes in terms of first-order closure or mixing coefficient cT as follows (see Panofsky and Dutton 1984):
i1520-0493-135-7-2610-e7
i1520-0493-135-7-2610-e8

There are equations comparable to Eqs. (1)(5) for the wind components as found in Liu et al. (1992). However, we do not include these in our set, in part because of the observational evidence that the momentum is not mixed, but also because we are able to obtain reasonably good estimates of the surface wind from our analysis, which is fundamental for the turbulent transfer of heat and moisture.

Equation (1) expresses the increase in temperature of the mixed layer in response to heating from below (turbulent transfer from ocean to air) and above [entrainment of warm air (potential temperature) above the mixed layer]. Equation (2) expresses the rate of growth of the mixed layer in response to entrainment and the imposed large-scale subsidence. Equation (3), in principle similar to Eq. (1), is the governing constraint for water vapor content in the mixed layer. Equations (4) and (5) express the temporal change in the jumps of temperature and water vapor, respectively, at the top of the mixed layer. Equations (4) and (5) follow from the temporal changes of temperature, vapor, and height in the presence of the assumed profiles; that is, they are derived from the geometry of the profiles. When the mixed layer is overlain by cloud (stratus), radiative heat transfer processes must be included (see Lilly 1968).

As will be shown in our experiments with these governing equations, the solutions are “weakly nonlinear.” That is, solutions resulting from perturbations to the control vector are well approximated by the first-order terms in the Taylor expansion about the base state. This has advantages in the sensitivity analysis. Namely, a change in the model output in response to a change in the control vector is accurately determined from first derivatives alone.

4. Base state and uncertainties in the control vector

Upper-air observations from Salvia are used to find the initial state and the parameters γθ, γq, and w (w discussed in 2c). The initial state is given by H0 = 250 m, θ0 = 10°C, σ0 = 1.5°C, q0 = 4 g kg−1, μ0 = −1.5 g kg−1, and Vs = 10 m s−1.The parameters are given by cT = 10−3, κ = 0.3, γθ = 3.3 × 10−3°C m−1, w = −1.0 cm s−1, and γq = −10−3 g kg−1 m−1, where the entrainment parameter κ and the turbulent transfer coefficient cT are specified in accordance with information found in the literature [Stage and Businger (1981) and Kondo (1975) for cT and Tennekes and Driedonks (1981) and Lilly (1987) for κ]. As discussed in these studies, it is difficult to precisely specify these empirical parameters. More important to our problem is the specification of uncertainties in these parameters about reasonable mean values. The base state is completed by specification of the boundary conditions along the trajectory, namely, the potential temperature at the sea surface (derived from the SST and atmospheric pressure along the trajectory). These boundary conditions vary from 16.5 to 24.0°C.

Nominal values of the uncertainties in the control vector follow: ΔH0 = ±5.0 × 10 m, Δθ0 = ±1.0°C, Δσ0 = ±5.0 × 10−1°C, Δq0 = ±5.0 × 10−1 g kg−1, Δμ0 = ±5.0 × 10−1 g kg−1 Δκ = ±1.0 × 10−1, Δγθ = ±2.0 × 10−4°C m−1, Δγq = ±5.0 × 10−4 g kg−1 m−1, Δw = ±5.0 × 10−1 cm s−1, Δθs = ±1.0°C, and ΔVs = ±2 m s−1. In association with the uncertainty in Δθs, there is an uncertainty in cT of ±1.0 × 10−4 (Kondo 1975; Stage and Businger 1981) and an average uncertainty in qs of 0.87 g kg−1.3

The term “uncertainty” deserves discourse. In regard to SST, it is reasonable to think of uncertainty as departure from truth in response to observational error—error in the instrument itself and/or error in the contamination of the radiances by cloud. Although absent in our study, it is not unusual to have the coastal waters blanketed with stratus cloud following outflow. In these cases, aged data from an earlier cloud-free time is used and this typically leads to error and discontinuities in the field as discussed in section 2a.

The parameters w, γθ, and γq are found from upper-air observations and are treated as constants. Yet, we know that these variables are not constant over the period of outflow. Thus, uncertainty for these variables can be viewed as natural variability during the event that is inconsistent with the model assumptions. And, of course, errors in the upper-air observations will also contribute to uncertainty in these parameters.

Akin to the SSTs, the initial conditions are associated with observational errors. However, it is also appropriate to include “errors of representation”—that is, errors that accrue from inadequate observational networks. Along the United States’s portion of the Gulf coast, the upper-air observations are separated by 300–400 km. Consequently, important smaller-scale detail cannot be incorporated into the initial state. Parameters such as cT and κ are fundamentally linked to the turbulence in the layer and therefore exhibit variability/uncertainty in response to the thermal forcing at the sea surface—especially the air–sea temperature difference in these thermally driven boundary layer situations.

In our experiments, we will consider two types of uncertainty—that associated with bias and that associated with unbiased randomness. Bias will imply a systematic uncertainty, either greater than or less than the true or correct mean value of the element, while randomness will imply random uncertainty about the mean value of the element. A combination of both types of uncertainty will also be considered in this study.

5. Tests with the analytic solution

a. Form of the solution

If water vapor and imposed large-scale vertical motion are not included in the mixed-layer system—that is, if we set w = 0 and if we remove Eqs. (3) and (5) from the set—an analytic solution to these equations can be found. We refer to this set of equations as the special case to distinguish it from the general case that does not possess an analytical solution and requires numerical integration to find the approximate solution. The analytic solution for the special case was found by Manton (1980), and later generalized by Driedonks (1982). The generality introduced by Driedonks rested on allowance for variability of heat input to the column. The special case bears strong mathematical resemblance to the more complete system, and there is advantage to explore errors in the numerical solution. Because we are interested in determining the change in model output w.r.t. changes in the control vector, knowledge of errors in these derivatives is relevant to our study. Further, the analytic solution can be used to determine the relative magnitudes of the first-order and second-order terms in the Taylor expansion of the output around the base state—in short, the degree of nonlinearity can be ascertained.

We write the analytic solution under the constraints listed above, where subscript 0, i.e., ( )0, refers to the initial conditions:
i1520-0493-135-7-2610-e9
i1520-0493-135-7-2610-e10
i1520-0493-135-7-2610-e11
where the heat input I(t), and the initial state parameter D0, are given by
i1520-0493-135-7-2610-e12
i1520-0493-135-7-2610-e13

The solution is valid when I(t) > D0 [see Eq. (9)]. This limitation arises from neglect of terms in the ratio (H0/H)κ−1, terms that are negligibly small shortly after initiation of the mixed-layer process (see Driedonks 1982, section 4). This condition is satisfied for t ≥ 1.1 h in our particular case.

b. Estimates of numerical errors in the sensitivity

From the analytic solution we can find the sensitivity, traditionally defined in terms of fractional change in the output of interest—call it p—to the fractional change in the element of the control vector—call it q. Then, the sensitivity of p to q, Rp,q, is given by
i1520-0493-135-7-2610-e14
The seven elements of the control vector are H0, θ0, σ0, κ, γθ, Vs, and θs. The exchange coefficient cT is a function of the sea–air temperature difference, θsθ = δθ,4 consequently, it is not an independent element of the control vector. In this analytic case, we assume that δθ remains constant; that is, potential temperature at the sea surface increases at the same rate as the mixed layer’s potential temperature. We assume this constant is 7°C, the average air–sea potential temperature difference during outflow for our case study. Because this difference only appears in combination with Vs and cT, we combine these variables as follows:
i1520-0493-135-7-2610-e15
Thus, the number of control variables is reduced from seven to six. Using this combination of control variables, I(t) = θ̂st. In Table 1, we display the analytic expressions for the sensitivity of θ(t) and H(t) to the six elements of the control vector. Without loss of generality, we confine our analysis to the temperature and height variables. The arguments we make regarding temperature and height apply to the temperature-jump variable.

Because the general equations do not possess an analytic solution, we plan to perform numerical integration of these equations via the fourth-order R–K scheme (Press et al. 1992). We test the goodness of this finite-difference method by comparing the numerical solution with the analytic solution for the special case. A small error in the solution does not guarantee small errors in the derivatives of the solution; that is, the structure of the solution in the space of the control vector will generally exhibit curvature for any nonlinear system. In view of the dynamic similarity between the special case and the general case, we are confident that these tests yield reliable estimates of errors in the numerical calculation of sensitivity for the general case.

The analytic solution over an 18-h period is shown in Table 2. The numerical solution is virtually identical to the analytic solution. Errors in the numerical evaluation of the sensitivity are displayed in Table 3. Here the errors are generally less than 5%. To calculate the numerical approximations to the sensitivity, the interval over which the derivative is calculated must be specified. Typically, the interval was taken to be a sizable fraction of the uncertainty in the elements of the control vector (section 4). The results are not overly sensitive to the size of the interval—an indication that variability of the solution about the base-state elements is “nearly” linear.

c. Linearity

The degree of linearity can be found in this analytic case since the second derivatives of the variables with respect to the elements of the control vector are known exactly. For the temperature, the Taylor series expansion about the base state, , takes the form
i1520-0493-135-7-2610-e16
where c′ is the perturbed control vector; c is the base-state control vector; θ is the gradient of θ w.r.t. the elements [the ∂p/∂q portion of the expression in Eq. (14)]; ψθ is the Hessian, a 6 × 6 symmetric matrix of second derivatives of θ w.r.t. the elements; and superscript T represents the transpose of a vector. The Taylor series expansions for H and σ follow directly from the form of Eq. (16) where ψH and ψσ are the corresponding Hessians. We find that when the perturbations (c′ − c) are given by the uncertainties shown earlier, the sum of all second-order terms are at least one order of magnitude less than the first-order-terms. The consequence of this result is that the sensitivities are directly proportional to the size of the perturbation/uncertainty. Thus, if the base state remains unchanged, sensitivity to another value of the uncertainty is found by simple multiplication of the earlier result.

d. Systematic uncertainty in the control vector

We assume the analytic solution given in Table 2 is the truth. We now ask the question: If our specification of the elements of the control vector is uncertain, how does this affect the model output? We will be interested in the model output at t = 18 h (the assumed termination of the outflow phase). Again, without loss of generality, we focus on the model output of temperature and height.

Table 4 shows the sensitivity of the forecasted height and temperature to elements of the control vector—changes in the outputs p (H and θ) in response to changes in the elements q (H0, θ0, . . . , θ̂s), the ∂p/∂q portion of the nondimensional sensitivity Rp,q that appears in Eq. (14). The uncertainties Δxi shown in column 3 of this table have been previously presented in section 4. Uncertainties are assumed positive to simplify interpretation of the results. The perturbation in θ̂s is 3.6 × 10−2°C m s−1, found by assuming that the perturbed control element θ̂s is the product of δθ (=8°C), cT (=1.1 × 10−3), and Vs (=12 m s−1).

In columns 4 and 6 of Table 4, we display uncertainties in the outputs that stem from uncertainties in each of the elements. For example, we show that an uncertainty of 50 m in the initial height of the mixed layer (H0) will lead to a biases of −8.0 m in height and −0.2°C in temperature at t = 18 h. A graphical presentation of these results is found in Fig. 10. Not surprisingly, the largest bias stems from uncertainly in θ̂s, the parameter that influences the turbulent flux through dependence on the wind speed, the air–sea temperature difference, and the mixing coefficient. The last row of Table 4, labeled “optimal,” places an upper limit on the output bias.

e. Random uncertainty in the control vector

Because random error is generally present in elements of the control vector, it is instructive to examine the probability density function (pdf) of the output under conditions of randomness. We will assume that the randomness in the elements is represented by a normal or Gaussian pdf, where the mean value is the true base-state value of the element, and where the standard deviations (square root of the variances) are the assumed uncertainties specified earlier in section 4.

We construct the pdf of the model output by performing a Monte Carlo experiment. The experiment is executed as follows:

  • Step 1: A set of elements of the control vector is found by randomly choosing values from the respective pdfs.

  • Step 2: The analytic solution is found for this particular set of elements. The model solution at t = 18 h is the output of interest.

  • Step 3: The model output (θ, H, and σ) is placed in a set of “bins,” where the ranges of these bins are established a priori.

  • Step 4: Return to step 1 and continue through step 3 for a total of 1000 model simulations.

  • Step 5: The relative number of model solutions in each bin constitutes approximations to the pdfs for the model variables.

When the elements of the control vector are random but unbiased, the pdf for the height of the mixed layer at t = 18 h is shown in the top panel of Fig. 11. If we systematically choose the wrong mean values of the entrainment parameter κ and lapse of temperature parameter γθ, then we expect biased results. We now ask the question: In the presence of systematically incorrect mean values of these two parameters, what are the consequences of including random error? We again examine this question by performing a Monte Carlo experiment. The only difference between this experiment and the one discussed above is that we now have different pdfs for κ and γθ. The mean values of these parameters are the systematically wrong values, 0.4 instead of 0.3 for κ, and 3.5 × 10−3 °C m−1 instead of 3.3 × 10−3 °C m−1 for γθ. The results of this experiment are shown in the lower panel of Fig. 11. The mean value of the height is increased by 61 m. This outcome could have been anticipated by examination of results found in Table 4 (rows 5 and 6—the rows showing results for κ and γθ). Here we note that a change in κ (by +0.1) results in a systematic change in H by 127 m. Further, a change in γθ (by +0.0002) results in a systematic change in H by −62 m. The sum of these changes is +65 m, close to the +61 m difference between mean values of the pdfs shown in Fig. 11. The nonlinearity of the dynamical constraints accounts for this difference of several meters.

6. Simulations with the general mixed-layer model

a. Validation

In our baseline numerical experiment with the general model, we choose a sampling interval of 36 km—a distance commensurate with a translation speed of 10 m s−1 for 1 h. Thus, over the 648-km trajectory (movement of the air column at 10 m s−1 for a period of 18 h), there are 19 sampling points—19 points where the SSTs are specified. Additionally, there are 11 other elements in the control vector (itemized at the beginning of section 4). Consequently, there is a total of 30 elements in the control vector for our general mixed-layer model.

We validate the prediction model against the inferred structure of the mixed layer as found from the soundings (profiles shown in Figs. 5 and 6). The results of this validation exercise are shown in Table 5. With the strong forcing governed by the air–sea temperature difference of ∼7°C, the growth of the layer and increases in temperature and vapor are guaranteed. The layer is a little too shallow (by ∼100 m) and a little too cool (by ∼1°C) at the terminal point, but the overall deepening and warming are creditable. The vapor forecast, both in the mixed layer and in the jump atop the layer, are especially well forecasted. The temperature-jump forecast is poor. And as stated earlier, the extreme increase in temperature jump is linked to the subsidence warming above the mixed layer—a process that is not included in the model.

b. Systematic uncertainty

We now examine the consequences of systematic uncertainty in the elements of the control vector. In Table 6, we have itemized the changes in the model output at t = 18 h in response to systematic bias in the initial conditions. Again, for ease of interpretation, we have assumed that the uncertainties Δxi are positive.

Examination of the entries in Table 6 indicate that uncertainties in the initial temperature of the mixed layer, θ0, strongly influences the output in all variables. If the temperature is systematically too large by 1°C, the output height drops by ∼100 m, the output temperature increases by ∼0.7°C, and the vapor increases by ∼0.2 g kg−1. With this initial temperature, the forcing (θsθ) decreases and the mixed layer grows more slowly. In turn, the vapor and heat are mixed through a shallower layer. Although there may be a tendency to believe that the mixed-layer properties at the outflow terminal point “forget” the initial conditions in this case of strong forcing at the air–sea interface, entries in Table 6 refute such a conjecture. The net effects (“sum”) of these perturbations are approximately −170 m in height, 1.0°C in temperature, and 1.0 g kg−1 in vapor.

In demonstration of the linearity of the system, we have shown results for uncertainties that differ by a factor of 2. These results are shown for the elements H0 and θ0. As expected for weakly nonlinear systems, the adjustments to outputs differ by factors that are approximately equal to 2.

Table 7 is the complement of Table 6; that is, the response of the output to incremental changes in the boundary conditions and other physical and empirical parameters. Because uncertainty in θs controls uncertainty in cT and qs, we introduce the symbol θ*s to represent the combined uncertainty in all three variables. In our numerical experiments, when θs is changed, corresponding changes are made in cT and qs. A visual display of these results is found in Fig. 12. For the assumed uncertainties (all positive and given in the second column of Table 7), the sensitivities are presented in an ordered fashion in Fig. 12, the most influential at the top of each display and the least influential at the bottom. The principal features of these results follow:

  1. A systematic positive bias in θ*s (and associated bias in cT and qs) leads to significant overestimate of the temperature, water vapor content, and height of the mixed layer. The same statement applies to the translation speed Vs where an increase in speed from 10 to 12 m s−1 implies a 20% increase in the fluxes [Eqs. (7) and (8)].

  2. Uncertainties in lapse rates (γθ and γq) have minor influence on the height.

  3. The height is strongly influenced by the large-scale vertical motion and θ*s. A positive bias in w leads to a too-deep mixed layer. In turn, this spreads the heating over too deep a layer and leads to a decrease in mean temperature. Although the too-deep mixed layer contributes to a dilution of the vapor (a decrease in q), there is a compensating influence related to the entrainment of dry air from above—weμ is less negative compared with the control when the vertical motion is increased. This stems from an increase in the temperature jump, σ, which reduces the entrainment velocity we.

  4. Overestimation of the entrainment parameter results in a warmer/drier mixed layer.

  5. The water vapor bias is most dependent on the uncertainty in the lapse of vapor above the mixed layer. An increase in γq (reduction of the lapse rate) contributes to moistening—the air entrained from above is moister in such an instance. And as might be expected, the rate of entrainment of the air (proportional to the empirical parameter κ) is also found to be an important factor. An overestimate of this parameter results in drying.

The sensitivity analysis demonstrates that it is possible to make an accurate forecast with compensatory/offsetting uncertainties in the elements of the control vector. For example, as seen by examination of Fig. 12, reasonably good moisture forecasts can be generated via a systematic positive bias in the SST that is offset by a comparable positive bias in the entrainment parameter. These biases, however, will lead to a systematic error in the forecasted height of the mixed layer. Even in this simple model, there are abundant ways to improve a poor forecast in one variable at the expense of degrading the forecast in another variable.

c. Random uncertainty

We have performed a Monte Carlo experiment with the general mixed-layer model. The mean values of the elements of the control vector are assumed to be correct, but uncertainties about the mean exist and their magnitudes are given in section 4. Results of these experiments are shown in Table 8. From the tabular values, it can be seen that the standard deviations about the means of these ensemble statistics are roughly half the value of the systematic shifts shown in Table 7 (the “Sum” row in this table). If we assume that these ensemble statistics are normally distributed, then only 2% of the ensemble members exhibit extremes that are comparable to the systematic shift of the variables (temperature, height, and vapor content) due to bias. In short, only rarely would unbiased 18-h forecasts exhibit outcomes comparable to those associated with the biased forecasts shown in Table 7.

7. Discussion

For the case we have studied, there is little doubt that the outflow phase of return flow is characterized by a thermodynamic structure close to that associated with the classic mixed-layer model. From the NCEP memorandum discussed earlier (Manikin 1998), there is also evidence that the operational model exhibits some of these characteristics. Specifically, there is a well-mixed temperature profile (constant θ), but a “poor shape of the dewpoint curve . . .” (Manikin 1998, p. 2). Essentially, the mixing of moisture is not commensurate with the mixing associated with the heat. This result begs the question: Is the latent heat flux from the operational model correct in the presence of an incorrect vertical distribution of the moisture?

To answer this question, it is necessary to verify the model-generated fluxes with measurements over the Gulf. A valuable source of information that can be used in such a verification comes from moored buoys maintained and operated by NDBC (Hamilton 1986). In the mid-1990s, several of the buoys were equipped with dewpoint sensors in addition to the standard instruments to measure wind speed and direction, SST, and air temperature (Breaker et al. 1998).

Measurements from an NDBC buoy in the northeast corner of the Gulf, 106 n mi west-northwest of Tampa, Florida, are shown in Fig. 13. (See Fig. 3 for the location of this buoy.) It is moored in water that is 54 m deep. The measurements shown in this figure are associated with a return-flow event that exhibited a bias similar to that discussed in the introduction—namely, the forecasted moisture return along the western Gulf was positively biased by 1–2 g kg−1. With strong steady winds of ∼20 kt (∼10 m s−1), the water most likely mixes to the ocean floor and becomes isothermal. This overturning contributes to a cooling of the surface water, but the principal reason for the SST drop is the loss of heat from the ocean via turbulent fluxes (see Lewis et al. 1997). Using the bulk formulas as found in Kondo (1975), we find the maximum sensible and latent heat fluxes in this instance to be ∼100 and ∼500 W m−2, respectively, giving a Bowen ratio of 0.2. These fluxes [proportional to the (qsq) and (TsT) curves in the bottom panel of Fig. 13] exhibit a rapid increase during the first day of the event, then become steady for nearly two days and finally diminish as the winds become easterly. The SST at this location drops by ∼2°C over the 3-day period (00/13 through 00/16). Thus, the frequency of updating the SSTs becomes an important issue. Certainly, the potential value of a coupled ocean–atmosphere model for return-flow events is evident.

At operational centers such as NCEP, time constraints do not allow us the luxury of producing a man–machine interactive SST product. Nevertheless, the value of such a product in return-flow situations is undeniable. The following question then looms: How do we augment the radiance-derived SST product to incorporate small-scale detail? On the optimistic side, buoy data are currently being incorporated into the SST product at NCEP. The operational SST analysis uses the latest 24-h set of in situ and satellite-derived SSTs. It is a global analysis on a grid that is 0.5° latitude by 0.5° longitude (approximately 55 km × 50 km over the Gulf). When this upgraded SST analysis was introduced in 2001, there was hope that it would lead to improved return-flow forecasts. Yet the bias remained, as retrospectively viewed:

The new SST product was clearly a major improvement and has certainly improved many aspects of model and human forecasts; it just wasn’t sufficient to significantly reduce the Eta’s moist bias with return flow events. (G. Manikin 2006, personal communication)

Intuitively, a coarse resolution of the SST field will lead to a positive bias in the moisture forecast. That is, the strong gradient of SST over the shelf water is smoothed in the presence of coarse resolution. The result is an increase in the SSTs over the shelf and a commensurate increase in the fluxes of moisture and heat.

Examination of the wind profiles from Salvia makes it clear that the momentum did not exhibit the mixing experienced by the moisture and heat. Ideally/theoretically, one expects momentum mixing in these unstable, penetrative convective boundary layers (Tennekes and Driedonks 1981). Indeed, the operational boundary layer model (the BMJ scheme at NCEP) generally yields these idealized results (Z. Janjic 2006, personal communication). Thus, at least in this case study, the operational forecast of wind in the marine layer would likely exhibit significant differences compared to the observations.

In discussions with researchers at NCEP (Z. Janjic and G. Manikin 2006, personal communication), hope for improved return-flow forecasts hinges on improved SST analysis and eventual incorporation of nonparameterized convective processes. The following statement assesses the current state of affairs at NCEP and provides a view of future plans:

We are actually running an experimental version of the WRF [Weather Research and Forecasting model] with 4.5 km resolution. In this version there is no parameterized convection, i.e., convection is simulated explicitly . . . although the WRF can handle the convection at this resolution without computational problems, and provide useful guidance for the forecasters, this still does not mean that the convection is represented with sufficient accuracy. We need still higher resolution for accurate representation of convection . . . we are now in a gray area between resolutions where we can simulate convection explicitly (on the order of 1 km), and resolutions for which convection parameterizations have been developed (on the order of 100 km). Namely, at upper single digit resolutions, the assumptions used to develop existing convective parameterizations generally cease to be valid. So far there is no satisfactory convection parameterizations on those scales. (Z. Janjic 2006, personal communication)

8. Conclusions

This research has aimed at uncovering the sources of biased errors in operational moisture forecasts of return flow over the Gulf of Mexico. By focusing on the outflow phase of the phenomenon, it has been shown that there is justification for the use of a low-order Lagrangian mixed-layer model. With the limited number of elements in the control vector of this model, a comprehensive analysis of the sensitivity study has been achieved. The most important elements, that is, those elements whose uncertainty is most likely to lead to biased forecasts, are the SSTs and the surface winds—the boundary conditions that directly impact the turbulent transfer of heat and vapor from sea to air. Even with accurate SSTs, it has been argued that it is expe- dient to compare operational model-generated fluxes with those calculated from the buoy data. To unravel the mystery that surrounds these biased moisture forecasts, we must continue to evaluate operational model output in light of all available observations and uncertainties in the model’s control vector, especially those uncertainties in the boundary layer and convective parameterizations. This is especially challenging in light of the absence of routine upper-air observations over the Gulf.

Acknowledgments

Awareness of systematic errors in the operational forecasts of return flow came from discussions with forecasters/researchers at the Storm Prediction Center (SPC) in Norman, Oklahoma, and the EMC in Camp Springs, Maryland. Comments from the following individuals were most helpful: Geoff Manikin and Zavisa Janjic (EMC), and Steve Weiss and Richard Thompson (SPC). A series of lectures by Doug Lilly on mixed-layer modeling and penetrative convection during the fall of 1987 at University of Oklahoma were enlightening. A debt of gratitude is owed to Lt. Cmdr. Gordon Garrett (USCG), captain of the Salvia, and his crew of 57. They, along with upper-air specialists Bob MacBeth (National Center for Atmospheric Research) and Thomas Hicks (National Weather Service, Southern Region), made it possible to collect the upper-air data used in this study. We are also indebted to Steve Baig, former research oceanographer at NOAA/NHC, for creating the man/machine SST product used in this study. We thank the National Geographic Society, particularly Eric Lindstrom, the Society’s senior map editor and map library director, for permission to reproduce the bathymetric chart of the Gulf of Mexico (found in the top portion of Fig. 4). Joan O’Bannon, graphics artist at National Severe Storms Laboratory, is credited with the production of electronic versions of figures in this paper, and Domagoj Podnar and Vicki Hall assisted in the electronic submission process. Informal reviews of the manuscript at an early stage by Bob Maddox and William Thompson, along with formal reviews by MWR-appointed reviewers, were most valuable.

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Fig. 1.
Fig. 1.

NCEP forecasts and verification of surface moisture return over the Gulf of Mexico. Forecasts are displayed on the right and verifications on the left.

Citation: Monthly Weather Review 135, 7; 10.1175/MWR3430.1

Fig. 2.
Fig. 2.

The trajectory of surface air that began its track at 1200 UTC 21 Feb 1988, near the mouth of the Mississippi River. This trajectory overlies the contours of sea surface temperature (°C). The dashed line is the continuation of the trajectory after the final upper-air observation was taken.

Citation: Monthly Weather Review 135, 7; 10.1175/MWR3430.1

Fig. 3.
Fig. 3.

Sea level pressure analyses and surface winds over the Gulf of Mexico on (left) 21 Feb and (right) 22 Feb 1988. The location of the NDBC buoy off Tampa Bay, FL, is indicated.

Citation: Monthly Weather Review 135, 7; 10.1175/MWR3430.1

Fig. 4.
Fig. 4.

(top) The bathymetry of the Gulf of Mexico where the depths of the ocean at several locations are given in meters. (Courtesy of the National Geographic Society.) (bottom) A composite map of sea surface temperatures (°C) over the Gulf of Mexico and Caribbean Sea during the one-week period, 20–26 Feb 1988.

Citation: Monthly Weather Review 135, 7; 10.1175/MWR3430.1

Fig. 5.
Fig. 5.

Profiles of potential temperature obtained from upper-air observations along the trajectory shown in Fig. 2. The times of observation are 1) 1138 UTC 21 Feb, 2) 1728 UTC 21 Feb, 3) 0005 UTC 22 Feb, 4) 0300 UTC 22 Feb, and 5) 0530 UTC 22 Feb.

Citation: Monthly Weather Review 135, 7; 10.1175/MWR3430.1

Fig. 6.
Fig. 6.

Profiles of specific humidity obtained from upper-air observations along the trajectory shown in Fig. 2. The humidity sensor on the dropsonde failed at 0300 UTC 22 Feb and thus profile 4 is absent.

Citation: Monthly Weather Review 135, 7; 10.1175/MWR3430.1

Fig. 7.
Fig. 7.

Visible imagery of cloud over the eastern Gulf of Mexico, Florida, and Cuba, at 1501 UTC 21 Feb 1988.

Citation: Monthly Weather Review 135, 7; 10.1175/MWR3430.1

Fig. 8.
Fig. 8.

Upper-air wind profiles collected along the low-level trajectory shown in Fig. 2.

Citation: Monthly Weather Review 135, 7; 10.1175/MWR3430.1

Fig. 9.
Fig. 9.

Schematic diagram depicting the structure of the idealized mixed layer. Symbols used for the variables and parameters are indicated.

Citation: Monthly Weather Review 135, 7; 10.1175/MWR3430.1

Fig. 10.
Fig. 10.

Bias in forecasts of (left) temperature and (right) mixed-layer height in response to uncertainty in the elements of the control vector (analytic case).

Citation: Monthly Weather Review 135, 7; 10.1175/MWR3430.1

Fig. 11.
Fig. 11.

Pdf for the height of the mixed layer at t = 18 h (analytic case). (top) Pdf when the elements of the control vector are unbiased. (bottom) Pdf when κ and γθ are biased.

Citation: Monthly Weather Review 135, 7; 10.1175/MWR3430.1

Fig. 12.
Fig. 12.

Pictorial representation of the sensitivities found in Table 7.

Citation: Monthly Weather Review 135, 7; 10.1175/MWR3430.1

Fig. 13.
Fig. 13.

Measurements from a moored buoy off the coast of Tampa Bay, FL, during a return-flow event in February 1999. Time is indicated along the top of each panel ranging from 0000 UTC 13 Feb (denoted by 00/13) to 0000 UTC 16 Feb (denoted by 00/16). Variables: sea surface temperature Ts (°C), air temperature T (°C) at the 3-m level, mixing ratio at sea level qs (g kg−1), and the mixing ratio q (g kg−1) at the 3-m level. The wind speed (kt) and direction are displayed between the two panels.

Citation: Monthly Weather Review 135, 7; 10.1175/MWR3430.1

Table 1.

Analytic expressions for the elements of the sensitivity vector.

Table 1.
Table 2.

Analytic solution to the mixed-layer equations for the special case.

Table 2.
Table 3.

Elements of the sensitivity vector—analytic solution and numerical solution for the special case.

Table 3.
Table 4.

Change to H and θ (analytic case) in response to incremental changes Δxi in the elements of the control vector. Sum refers to the cumulative sum of the first-order terms in the presence of positive increments. Optimal refers to the cumulative sum when the sign of the increment is chosen such that the first-order terms are all positive; i.e., if the derivative is negative, the increment is taken to be negative while a positive derivative is associated with a positive increment.

Table 4.
Table 5.

Evolution of the meteorological variables in the mixed-layer model. The simulation began at 1200 UTC 21 Feb (t = 0). Verification data from the upper-air soundings are found in the parentheses at three times: 1728 UTC 21 Feb (t ≅ 5.5 h); 0005 UTC 22 Feb (t ≅ 12 h); and 0530 UTC 22 Feb (t ≅ 17.5 h).

Table 5.
Table 6.

Changes to H, θ, and q in response to incremental changes Δxi in the initial conditions. The prime denotes the perturbed state while the overbar denotes the base state. Sum and optimal are in accord with the description found in Table 4. Only the top rows of H0 and θ0 (increments of 50 m and 1°C, respectively) are used to calculate these sums.

Table 6.
Table 7.

Changes to H, θ, and q in response to incremental changes Δxi in the boundary condition and other physical parameters. The prime denotes the perturbed state while the bar denotes the base state. Sum and optimal are in accord with the description found in Table 4.

Table 7.
Table 8.

Monte Carlo experiment with the general mixed-layer equations. These statistics of the model output at t = 18 h are the result of 1000 simulations where the elements of the control vector are normally distributed. The means and standard deviations of the elements of the control vector are in accord with the means and uncertainties (set equal to the standard deviations) presented in section 4.

Table 8.

1

Hereafter referred to as NCEP.

2

Bracketed information, [. . .], has been inserted by the author.

3

The uncertainty in qs is found by assuming saturation at the sea surface and by using Tetans formula in conjunction with the definition of specific humidity (Saucier 1955). The stated uncertainty is the average over the trajectory.

4

We use the symbol δθ to distinguish it from uncertainty, Δθ0 and Δθs.

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  • Fig. 1.

    NCEP forecasts and verification of surface moisture return over the Gulf of Mexico. Forecasts are displayed on the right and verifications on the left.

  • Fig. 2.

    The trajectory of surface air that began its track at 1200 UTC 21 Feb 1988, near the mouth of the Mississippi River. This trajectory overlies the contours of sea surface temperature (°C). The dashed line is the continuation of the trajectory after the final upper-air observation was taken.

  • Fig. 3.

    Sea level pressure analyses and surface winds over the Gulf of Mexico on (left) 21 Feb and (right) 22 Feb 1988. The location of the NDBC buoy off Tampa Bay, FL, is indicated.

  • Fig. 4.

    (top) The bathymetry of the Gulf of Mexico where the depths of the ocean at several locations are given in meters. (Courtesy of the National Geographic Society.) (bottom) A composite map of sea surface temperatures (°C) over the Gulf of Mexico and Caribbean Sea during the one-week period, 20–26 Feb 1988.

  • Fig. 5.

    Profiles of potential temperature obtained from upper-air observations along the trajectory shown in Fig. 2. The times of observation are 1) 1138 UTC 21 Feb, 2) 1728 UTC 21 Feb, 3) 0005 UTC 22 Feb, 4) 0300 UTC 22 Feb, and 5) 0530 UTC 22 Feb.

  • Fig. 6.

    Profiles of specific humidity obtained from upper-air observations along the trajectory shown in Fig. 2. The humidity sensor on the dropsonde failed at 0300 UTC 22 Feb and thus profile 4 is absent.

  • Fig. 7.

    Visible imagery of cloud over the eastern Gulf of Mexico, Florida, and Cuba, at 1501 UTC 21 Feb 1988.

  • Fig. 8.

    Upper-air wind profiles collected along the low-level trajectory shown in Fig. 2.

  • Fig. 9.

    Schematic diagram depicting the structure of the idealized mixed layer. Symbols used for the variables and parameters are indicated.

  • Fig. 10.

    Bias in forecasts of (left) temperature and (right) mixed-layer height in response to uncertainty in the elements of the control vector (analytic case).

  • Fig. 11.

    Pdf for the height of the mixed layer at t = 18 h (analytic case). (top) Pdf when the elements of the control vector are unbiased. (bottom) Pdf when κ and γθ are biased.

  • Fig. 12.

    Pictorial representation of the sensitivities found in Table 7.

  • Fig. 13.

    Measurements from a moored buoy off the coast of Tampa Bay, FL, during a return-flow event in February 1999. Time is indicated along the top of each panel ranging from 0000 UTC 13 Feb (denoted by 00/13) to 0000 UTC 16 Feb (denoted by 00/16). Variables: sea surface temperature Ts (°C), air temperature T (°C) at the 3-m level, mixing ratio at sea level qs (g kg−1), and the mixing ratio q (g kg−1) at the 3-m level. The wind speed (kt) and direction are displayed between the two panels.

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