#  A Brief History of Prompt: Leveraging Language Models.

Golam Md Muktadir1  1Computer Science and Engineering, University of
California, Santa Cruz
<EMAIL_ADDRESS>

###### Abstract

This paper presents a comprehensive exploration of the evolution of prompt
engineering and generation in the field of natural language processing (NLP).
Starting from the early language models and information retrieval systems, we
trace the key developments that have shaped prompt engineering over the years.
The introduction of attention mechanisms in 2015 revolutionized language
understanding, leading to advancements in controllability and context-
awareness. Subsequent breakthroughs in reinforcement learning techniques
further enhanced prompt engineering, addressing issues like exposure bias and
biases in generated text. We examine the significant contributions in 2018 and
2019, focusing on fine-tuning strategies, control codes, and template-based
generation. The paper also discusses the growing importance of fairness,
human-AI collaboration, and low-resource adaptation. In 2020 and 2021,
contextual prompting and transfer learning gained prominence, while 2022 and
2023 witnessed the emergence of advanced techniques like unsupervised pre-
training and novel reward shaping. Throughout the paper, we reference specific
research studies that exemplify the impact of various developments on prompt
engineering. The journey of prompt engineering continues, with ethical
considerations being paramount for the responsible and inclusive future of AI
systems.

###### Index Terms:

Prompt Engineering, Prompt Generation, Information Retrieval, Large Language
Models, Language Queries

##  I Introduction

Prompt #1: You are a scholar in machine learning and language models. I am
writing a paper on the history of prompt engineering and generation. Can you
give me a timeline for prompt engineering evolution? (We used this timeline to
create prompts for each section later)

Prompt #2: Write the introduction of this paper. Emphasize that this paper
focuses on how language prompts and queries have been used so far.

Natural Language Processing (NLP) has witnessed remarkable advancements over
the years, revolutionizing how we interact with machines and bridging the gap
between human language and artificial intelligence. One critical aspect that
has played a pivotal role in shaping the capabilities of NLP systems is the
design and usage of language prompts and queries. Recently, researchers and
practitioners have explored various techniques to harness the full potential
of language models, leveraging the power of prompts to guide, control, and
tailor the output of these sophisticated AI systems.

This paper embarks on a retrospective journey, offering a comprehensive review
of the captivating history of language prompts and queries in NLP. From its
nascent stages, where prompts were merely a means of input for rudimentary
language models, to its current state as an influential component in modern
deep learning models, we aim to unravel the captivating story behind this
remarkable facet of NLP research.

The inception of this narrative can be traced back to the early language
models of the pre-2010 era. These rudimentary models, such as n-grams and
statistical language models, laid the groundwork for understanding the concept
of prompt engineering - a concept that would become indispensable in the
future. While the terminology of ”prompt engineering” had not yet emerged, the
seeds of its significance were planted in information retrieval systems, where
queries were employed as inputs to retrieve relevant information from vast
datasets.

As the decade progressed, the 2010s witnessed a seismic shift in NLP, brought
about by the advent of neural networks. Groundbreaking innovations such as
Word2Vec, introduced in 2013 by Mikolov et al., paved the way for capturing
semantic relationships between words. Subsequently, the revolutionary
Sequence-to-Sequence (Seq2Seq) model, presented by Sutskever et al. in 2014,
unlocked the potential for machine translation through its encoder-decoder
architecture.

The transformational power of attention mechanisms came into the spotlight in
2015 with the release of the Transformer model, as proposed by Vaswani et al.
The attention mechanism allowed models to process and understand context with
unprecedented efficiency, facilitating long-range dependencies in sequence
data. This seminal development formed the cornerstone for future advances in
prompt engineering.

By 2017, the NLP community began to explore reinforcement learning techniques
for language generation. Researchers like Ranzato et al. and Li et al.
employed reinforcement learning to fine-tune language models based on external
reward signals. This pivotal step not only enhanced fluency and relevance in
generated text but also sowed the seeds for prompt engineering to emerge as a
key factor in guiding the model’s behavior and generating more contextually
relevant responses.

In 2018, a revolutionary breakthrough emerged with the introduction of BERT
(Bidirectional Encoder Representations from Transformers). Developed by Devlin
et al., BERT showcased the remarkable potential of pre-training language
models on vast corpora, followed by fine-tuning for specific tasks. This
epoch-making moment brought prompt engineering to the forefront, demonstrating
its power in leveraging pre-trained models for diverse NLP applications.

The late 2010s and early 2020s marked an era of innovation in prompt
engineering, as researchers explored control codes and template-based
generation methods. These techniques allowed for more targeted and controlled
text output, enabling researchers to guide language models towards specific
behaviors and desirable outcomes.

The advent of massive language models like GPT-3 in the year 2020 redefined
the landscape of NLP, elevating prompt engineering to new heights. With an
astonishing 175 billion parameters, GPT-3 showcased unparalleled language
generation capabilities, sparking widespread interest in prompt design and its
profound impact on fine-tuning language models for specific domains.

As the journey through time continues, we delve deeper into the history of
prompt engineering, exploring the rich tapestry of methodologies, milestones,
and applications that have shaped the course of NLP. By understanding this
evolution, we gain valuable insights into the challenges conquered, the
accomplishments celebrated, and the promising future that lies ahead for this
ever-evolving field of language prompts and queries in NLP.

##  II Prehistoric Prompting: Pre NN-Era

Prompt #3: Now generate history of prompting or querying in early language
models an information retrieval

Before the advent of neural networks and modern deep learning models, early
language models and information retrieval systems laid the foundation for the
concept of prompting or querying in natural language processing. Although not
explicitly referred to as ”prompting” or ”querying” at the time, these early
approaches formed the basis for how researchers and systems interacted with
textual data and sought relevant information.

###  II-A Early Language Models:

Early language models, such as n-grams and statistical language models,
emerged in the early days of computational linguistics. In these models, text
was processed as a sequence of words or characters, and the probability of the
next word was predicted based on the occurrence frequencies of n-grams
(sequences of n words). While these models did not involve explicit user
prompts or queries, the process of predicting the next word in a sequence can
be considered an implicit form of prompting, where the previous words serve as
input guiding the model’s prediction.

###  II-B Information Retrieval Systems:

Information retrieval systems have a long history predating modern NLP
techniques and were primarily used to retrieve relevant documents from large
collections based on user queries. Early information retrieval systems
utilized keyword-based search, where users provided queries consisting of
specific words or phrases, and the system returned documents containing those
keywords. These queries often required careful phrasing and choice of keywords
to obtain relevant results, effectively serving as explicit prompts to
retrieve desired information.

###  II-C 1960s-1970s: Early Natural Language Interfaces:

In the 1960s and 1970s, early attempts were made to develop natural language
interfaces for information retrieval systems. Researchers explored methods to
process user queries in natural language and transform them into suitable
search queries for information retrieval systems. These early systems provided
rudimentary prompting capabilities, allowing users to interact with the system
using natural language queries, albeit with limited understanding and
accuracy.

###  II-D 1980s-1990s: Rule-Based Approaches:

In the 1980s and 1990s, rule-based approaches were developed to enable more
sophisticated interaction with information retrieval systems. Researchers
designed complex query languages that allowed users to express search criteria
in a more structured and flexible manner. Users could specify constraints and
relationships between search terms, effectively prompting the system to narrow
down search results based on user preferences.

###  II-E 1990s-2010: Advances in Natural Language Processing:

The field of natural language processing witnessed significant advancements,
especially with the introduction of machine learning techniques and neural
networks. Modern information retrieval systems, search engines, and question-
answering systems leverage sophisticated language models and deep learning
techniques to interpret user queries and provide more accurate and
contextually relevant results. These systems employ explicit prompting through
user input and implicitly leverage the context within queries to optimize
search results and understand user intent better.

The historical development of prompting and querying in early language models
and information retrieval systems laid the groundwork for modern NLP
approaches that now utilize neural networks and large-scale language models
for more powerful and context-aware language understanding and generation. The
evolution of prompt engineering and querying continues to progress, with
ongoing research and innovations propelling the field further towards new
frontiers in natural language processing.

##  III History of NLP between 2010 and 2015: The Pre-Attention Mechanism Era

Prompt #4: Now write the history between 2010 and 2015 before attention
mechanism was invented

The period between 2010 and 2015 was marked by significant strides in Natural
Language Processing (NLP), laying the groundwork for revolutionary
advancements that would follow in subsequent years. During this era,
researchers explored various techniques to enhance language understanding and
generation, all leading up to the eventual introduction of the attention
mechanism, which would become a pivotal development in NLP. Here’s an overview
of the key developments during this period:

###  III-A Neural Network Renaissance (2010-2013):

NLP research saw a resurgence of interest in neural network-based approaches,
popularly known as the ”neural network renaissance.” Word embeddings gained
prominence as an effective way to represent words in continuous vector spaces.
The Word2Vec model proposed by Mikolov et al. in their paper ”Efficient
Estimation of Word Representations in Vector Space” (2013) was a breakthrough
in generating high-quality word embeddings. Researchers realized that these
continuous word representations allowed models to capture semantic
relationships between words, which led to better performance on various NLP
tasks.

###  III-B The Rise of Seq2Seq Models (2014):

In 2014, Sutskever et al. introduced the Sequence-to-Sequence (Seq2Seq) model,
which is an encoder-decoder architecture powered by recurrent neural networks
(RNNs). Seq2Seq models revolutionized machine translation by effectively
learning to encode the input sentence into a fixed-length vector (the
”context” or ”thought”), which is then used to generate the output sentence.
This architecture marked a shift towards more context-aware language modeling,
with the ability to process variable-length sequences and produce coherent and
contextually appropriate responses.

###  III-C The Emergence of Recursive Neural Networks (RNNs) and Long Short-
Term Memory (LSTM) (2010-2015):

Recurrent Neural Networks (RNNs) gained popularity as a means to process
sequential data, including natural language. LSTM, proposed by Hochreiter and
Schmidhuber in their paper ”Long Short-Term Memory” (1997) but gaining more
traction in the early 2010s, was a significant advancement over traditional
RNNs. LSTM addressed the vanishing gradient problem, allowing for the
successful training of deeper networks. LSTM-based models offered better long-
term dependency handling, making them suitable for tasks where maintaining
context over longer sequences was crucial.

###  III-D Limitations of Early Approaches:

Despite the progress made, early language models faced challenges in handling
long-range dependencies and maintaining context over lengthy sentences. RNNs,
while capable of capturing sequential information, suffered from vanishing or
exploding gradient problems during training, limiting their effectiveness in
understanding long sequences.

###  III-E Pre-Attention Mechanism Approaches:

Prior to the introduction of the attention mechanism, researchers devised
various methods to tackle the limitations of existing models. Techniques such
as Recursive Neural Tensor Networks (RNTNs) and Convolutional Neural Networks
(CNNs) were employed to capture hierarchical and compositional structures
within text. While these approaches showed promise, they still struggled to
effectively handle longer sequences and complex dependencies.

The period between 2010 and 2015 can be characterized as a time of
exploration, where researchers experimented with various neural network
architectures and representation techniques to improve language modeling and
generation. The advancements during this era laid the foundation for the
groundbreaking introduction of the attention mechanism in 2015, which would
revolutionize NLP by enabling more efficient context understanding and long-
range dependency handling in language models.

##  IV The Advent of Attention Mechanism: A Game-Changer for Prompt
Engineering in 2015

##  V 2015: Birth of the Transformer

Prompt #5: now write a section on how attention mechanism changed the future
of prompt engineering in 2015

In 2015, the field of Natural Language Processing (NLP) witnessed a
groundbreaking development that would forever change the landscape of prompt
engineering – the introduction of the attention mechanism. Proposed by Vaswani
et al. in the seminal paper ”Attention is All You Need”, [1], the attention
mechanism revolutionized the way language models processed input data,
enabling them to focus on relevant information and establish meaningful
contextual relationships. This transformative breakthrough had a profound
impact on prompt engineering, leading to improved language understanding,
better controllability, and enhanced performance across various NLP tasks.

###  V-A Contextual Understanding and Long-Range Dependencies:

The attention mechanism fundamentally changed how language models processed
sequences by allowing them to pay varying degrees of attention to different
parts of the input. Unlike traditional approaches that treated each word in
the sequence equally, attention-equipped models could dynamically assign
weights to each word based on its relevance to the context. This enabled the
model to establish meaningful relationships between words across long
sequences, overcoming the limitations of earlier methods and greatly improving
the contextual understanding of prompts.

###  V-B Better Prompt-Model Interaction:

With the attention mechanism, prompt engineering took on a new dimension as
the models could now focus on specific parts of the prompt that mattered most
for generating the desired output. Attention mechanisms facilitated an
interactive process, where the model could attend to important aspects of the
prompt, query, or context, refining its responses accordingly. This newfound
flexibility allowed prompt designers to craft more informative, targeted, and
nuanced queries, effectively guiding the model’s behavior and achieving more
desirable outcomes.

###  V-C Enhanced Controllability and Interpretability:

The attention mechanism empowered prompt engineers to exert finer control over
the generated text by explicitly specifying relevant portions of the input
that should receive heightened attention. By directing the model’s attention,
prompt engineers could guide the model’s focus and influence the style,
content, and tone of the generated output. Moreover, attention weights could
be interpreted and visualized, providing valuable insights into the decision-
making process of the language model, enhancing transparency, and aiding in
bias mitigation.

###  V-D Applications in Machine Translation and Language Generation:

The attention mechanism significantly improved machine translation systems by
allowing the model to focus on relevant source words when generating target
translations. In language generation tasks, such as text summarization and
question answering, attention mechanisms helped the model attend to crucial
context, leading to more coherent and contextually appropriate responses.

###  V-E Multi-Head Attention and Transfer Learning:

The concept of multi-head attention, introduced alongside the attention
mechanism, enabled models to attend to multiple aspects of the input
simultaneously, leading to improved performance and flexibility. The attention
mechanism also became instrumental in transfer learning. Pre-trained models
could be fine-tuned using prompt engineering to adapt to specific tasks
effectively, leveraging their context-awareness to improve performance on
various domains.

The introduction of the attention mechanism in 2015 represented a paradigm
shift in prompt engineering, propelling the field into a new era of language
understanding and generation. By empowering language models to focus on
relevant information and establish context-dependent relationships, the
attention mechanism unlocked new possibilities for fine-tuning models,
interpreting their decisions, and achieving higher levels of controllability
in natural language processing. As the attention mechanism continued to
evolve, researchers and practitioners explored new frontiers in prompt
engineering, opening the door to even more sophisticated and context-aware AI
systems.

##  VI The second wave in 2017: Rise of RL

Prompt #6: now discuss how the advent of reinforcement learning techniques in
2017 changed the prompt engineering

The introduction of reinforcement learning (RL) techniques in 2017 brought
another transformative wave to the field of prompt engineering. Reinforcement
learning involves training an agent (in this case, a language model) to take
actions in an environment (generate text) to maximize a reward signal. The use
of RL in conjunction with prompt engineering led to significant advancements
in fine-tuning language models, improving fluency, relevance, and
controllability of generated text, [2]. Here’s how the advent of RL techniques
changed the landscape of prompt engineering:

###  VI-A Improving Fluency and Relevance:

Reinforcement learning allowed prompt engineers to define appropriate reward
signals that could incentivize the language model to generate more fluent and
contextually relevant responses. Traditional supervised fine-tuning using
maximum likelihood estimation (MLE) often led to models that were overly
conservative and lacked creativity, but RL opened up possibilities for more
exploratory behavior. With RL, language models could explore the space of
potential responses, learning from their own generated samples and adjusting
their behavior based on the received reward signal, leading to more fluent and
contextually appropriate language generation, [3, 4].

###  VI-B Addressing Exposure Bias:

One significant challenge in language model training was exposure bias, where
a model is trained on teacher-forced input during training but experiences a
discrepancy during inference, often resulting in a gap between training and
testing performance. RL helped mitigate exposure bias by enabling models to
sample from their own predictions during training, aligning the training and
inference process more closely, [5, 6]. Prompt engineers could design reward
functions to encourage self-correcting behavior, leading to more consistent
and robust language generation during deployment.

###  VI-C Controlling Model Behavior through Reward Shaping:

Moreover, prompt engineers could use RL’s reward shaping to encourage specific
behaviors in the language model. By defining custom reward functions, prompt
engineers could guide the model to generate responses that adhered to desired
criteria, such as maintaining a specific tone, style, or level of formality,
[7]. This controllability allowed language models to be tailored for specific
applications, ensuring the generated content aligned with domain-specific
requirements.

###  VI-D Addressing Biases in Language Models:

Another important aspect was addressing biases in language models.
Reinforcement learning provided a means to mitigate biases in language models
by using carefully designed reward functions. By penalizing biased responses
or rewarding unbiased behavior, prompt engineers could encourage the model to
produce more equitable and unbiased language generation, contributing to
fairer AI systems, [8, 9].

###  VI-E Advancements in Task-Specific Prompt Engineering:

RL-powered prompt engineering also opened up avenues for researchers to
optimize language models for task-specific objectives. By defining tasks as
reinforcement learning problems and designing corresponding reward functions,
language models could be fine-tuned for applications such as text
summarization, dialogue generation, and question answering, leading to state-
of-the-art performance in various NLP tasks.

###  VI-F Challenges in Reward Design and Exploration:

However, while RL provided powerful tools for prompt engineering, designing
effective reward functions remained a challenging task. The choice of reward
functions could strongly influence the behavior of the model, requiring
careful consideration to ensure desired outcomes. Additionally, exploration-
exploitation trade-offs in RL training required balancing novelty and
diversity in generated text with the need for generating contextually
appropriate responses.

Overall, the incorporation of reinforcement learning techniques in prompt
engineering brought about a new level of controllability and performance to
language models. By shaping model behavior through reward functions and
addressing issues like exposure bias and biases in generated text, RL-powered
prompt engineering advanced the capabilities of language models, making them
more powerful, context-aware, and adaptable for a wide range of NLP
applications.

##  VII The third wave 2018: the rise of Transfomers

Prompt #6: Now write the section on how research in 2018 shaped prompt
engineering

The year 2018 was a pivotal period in the evolution of prompt engineering,
witnessing transformative breakthroughs that propelled the field to new
heights. Researchers explored innovative approaches to leverage pre-trained
language models, making use of transfer learning and fine-tuning techniques.
This year marked the rise of BERT (Bidirectional Encoder Representations from
Transformers), [10], which revolutionized NLP and significantly impacted
prompt engineering. Here’s an overview of how research in 2018 shaped the
landscape of prompt engineering:

###  VII-A The Emergence of BERT and Pre-training:

BERT, introduced by Devlin et al. in 2018, was a major milestone in transfer
learning for NLP. The model was pre-trained on a large corpus of unlabeled
text, allowing it to learn powerful language representations. BERT’s
architecture, based on the Transformer model with a bidirectional encoder,
captured contextual information from both left and right contexts, leading to
deeper and richer language understanding. The pre-training paradigm enabled
prompt engineers to leverage BERT’s knowledge and context-awareness while
fine-tuning for specific tasks, revolutionizing prompt engineering by
providing a starting point for more specialized models, [11].

###  VII-B Transfer Learning and Fine-tuning:

BERT popularized the concept of transfer learning in NLP. Researchers realized
that pre-training a language model on a vast corpus enabled it to capture
general linguistic patterns and context. Fine-tuning allowed prompt engineers
to adapt pre-trained models to specific downstream tasks with minimal
additional training data, [12, 13]. This transfer learning paradigm
drastically reduced the need for large task-specific datasets, making prompt
engineering more practical and effective.

###  VII-C Task-Specific Prompt Engineering with BERT:

Task-specific prompt engineering with BERT became prevalent in 2019. Prompt
engineers began utilizing BERT for a wide range of NLP tasks, such as
sentiment analysis, named entity recognition [14], and question answering,
among others. Task-specific prompt engineering involved providing the model
with carefully designed inputs, such as question-context pairs or masked
sentences, to guide its behavior and improve task performance. By fine-tuning
BERT on task-specific data, researchers achieved state-of-the-art results in
various NLP benchmarks, showcasing the power of prompt engineering with pre-
trained models.

###  VII-D Masked Language Model (MLM) and Cloze-Style Prompts:

BERT’s pre-training involved a masked language model (MLM) objective, where
random words in the input text were masked, and the model was tasked with
predicting the masked words, [15]. This MLM pre-training opened up new
possibilities for prompt engineering, as researchers could use cloze-style
prompts (with masked words) to guide the model to fill in the missing
information, generating coherent and contextually appropriate responses.

###  VII-E Contextualized Word Embeddings (ELMo):

Another influential development in 2018 was the introduction of ELMo
(Embeddings from Language Models) by Peters et al. ELMo generated
contextualized word embeddings by training bidirectional LSTM language models
on large corpora, [16]. ELMo embeddings allowed for richer word
representations that captured different meanings of a word in different
contexts, contributing to more sophisticated prompt engineering strategies.

###  VII-F Unsupervised and Semi-Supervised Prompt Engineering:

BERT’s ability to capture linguistic patterns from unlabeled data allowed for
unsupervised and semi-supervised prompt engineering approaches. Researchers
explored methods to fine-tune BERT on limited labeled data, using unsupervised
techniques to leverage abundant unlabeled data for improved task performance.

The research breakthroughs of 2018, centered around BERT and pre-training
techniques, profoundly shaped prompt engineering. Leveraging transfer learning
and fine-tuning, prompt engineers could tailor language models to specific
tasks with remarkable efficiency. The rise of BERT and contextualized
embeddings opened up new avenues for designing cloze-style prompts and
leveraging large amounts of unlabeled data for prompt engineering. These
advancements propelled the field of prompt engineering into a new era of
efficiency, adaptability, and state-of-the-art performance in NLP
applications.

##  VIII 2019: The Year of Control

Prompt #7: (a) Now write the section on developments in prompt engineering in
2019. (b) Now can you rewrite the section on developments in prompt
engineering in 2019? Please organize your thoughts in paragraphs instead of
bullet points.

The year 2019 saw remarkable progress in the field of prompt engineering, with
a strong focus on improving the controllability, interpretability, and
targeted generation of language models. Researchers explored novel techniques
to guide language models more effectively, shaping their behavior and
responses to suit specific requirements. This period witnessed advancements in
control codes, template-based generation, and innovations in fine-tuning
strategies, ushering in an era of more precise and context-aware language
generation. Here’s an overview of the key developments in prompt engineering
in 2019:

###  VIII-A Control Codes and Conditioning:

One of the key developments in prompt engineering in 2019 was the
incorporation of control codes into language model inputs. These control codes
are special tokens or markers added to the prompt, indicating desired
attributes, styles, or behavior in the generated text. By conditioning the
language model on control codes, researchers could guide it to produce content
adhering to specific criteria, such as sentiment, formality, or language
style. This controllability allowed for more targeted and contextually
appropriate responses, empowering prompt engineers to tailor language
generation to various use cases.

###  VIII-B Template-Based Generation:

Template-based generation also gained prominence during this period. Prompt
engineers designed prompts in the form of templates, with placeholders for
dynamic content. By providing specific values for the placeholders,
researchers ensured that the generated output followed the structure and
format defined in the template. Template-based approaches enabled more
structured and controlled text generation, making them valuable in
applications where precise and consistent responses were essential.

###  VIII-C Reinforcement Learning for Improved Controllability:

Advancements in reinforcement learning techniques further improved the
controllability of prompt-engineered language models. By refining reward
functions, prompt engineers could encourage the model to produce more
desirable responses, reducing biases and generating content that aligned
better with user preferences. Reinforcement learning played a crucial role in
refining prompt engineering strategies, enabling AI systems to learn from
human preferences and judgments.

###  VIII-D De-biasing Strategies:

Addressing biases in language models remained a critical focus in prompt
engineering in 2019. Researchers explored methods to de-bias prompt inputs and
mitigate potential biases present in the training data. Carefully crafting
prompts that avoid biased language or specifying fairness-related control
codes aimed to generate more equitable and unbiased language output.

Human-AI collaboration also became a significant aspect of prompt engineering.
In some cases, human-in-the-loop approaches were employed, where human-
generated responses served as reward signals for reinforcement learning. This
allowed the model to learn from human preferences and judgments, enhancing the
quality and relevance of generated content.

###  VIII-E Adapting to Low-Resource Languages:

Prompt engineering extended its impact to low-resource languages, where fine-
tuning large language models might be challenging due to limited training
data. Researchers explored methods to leverage transfer learning and
unsupervised pre-training, adapting prompt engineering techniques to address
the specific challenges of low-resource settings.

###  VIII-F Contextual Prompting and Dynamic Response Generation:

Furthermore, contextual prompting emerged as a powerful approach. Using
preceding context or user interactions as prompts allowed language models to
provide more dynamic and interactive conversation generation. Incorporating
contextual information enabled the model to provide coherent and contextually
appropriate responses, enhancing the overall user experience.

The developments in prompt engineering in 2019 propelled the field towards
enhanced controllability, targeted generation, and reduced biases in language
models. The integration of control codes, template-based generation, and
reinforcement learning techniques brought greater precision to prompt
engineering, aligning the generated text more closely with the desired
outputs. The exploration of human-AI collaboration and de-biasing strategies
marked significant steps towards responsible and fair AI systems. As the field
advanced, researchers continued to refine prompt engineering approaches,
laying the groundwork for future breakthroughs in making AI systems more
adaptable, interpretable, and human-centric.

##  IX 2020-2021: The rise of LLMs

Prompt #8: (a) now write the section for 2020 and 2021 in prompt engineering
(b) now rewrite the section for 2020 and 2021 in prompt engineering? Please
organize your thoughts in paragraphs instead of bullet points

The years 2020 and 2021 marked a period of unprecedented progress in prompt
engineering, largely driven by advancements in large-scale language models and
the democratization of AI technology. With the release of models like GPT-3
and innovations in prompt design, researchers and developers alike explored
new frontiers in controllability, interpretability, and domain adaptation.
These years witnessed the rise of massive language models, diversification of
prompt formats, and a growing emphasis on addressing ethical concerns. Here’s
an overview of the key developments in prompt engineering during this period:

###  IX-A The Age of Massive Language Models:

One of the defining features of this period was the emergence of massive
language models, [17, 18], exemplified by models like GPT-3 developed by
OpenAI. With its impressive 175 billion parameters, GPT-3 demonstrated
extraordinary language generation capabilities and found applications across
diverse domains. Prompt engineering techniques allowed researchers to leverage
these large models’ contextual understanding and capabilities, making them
more versatile and contextually intelligent in generating content across
various tasks and domains.

###  IX-B Advancements in Prompt Format and Style:

Advancements in prompt format and style were another key focus during 2020 and
2021. Researchers explored novel ways to design prompts, enabling more
targeted and controlled text generation. By utilizing different prompt types,
such as completions, instructions, or role-playing scenarios, prompt engineers
could guide models to produce specific styles, tones, or perspectives, [19,
20]. Creative prompt design empowered developers to create engaging and
contextually appropriate language generation systems, tailored to suit diverse
use cases and applications.

###  IX-C Domain Adaptation and Fine-Tuning:

Furthermore, domain adaptation and fine-tuning became integral to prompt
engineering in 2020 and 2021, [21, 22]. The widespread availability of pre-
trained language models allowed for more efficient adaptation to specific
domains and tasks. By fine-tuning models on smaller, task-specific datasets,
researchers achieved domain-specific performance without the need for training
massive models from scratch. Fine-tuning techniques made prompt engineering
more practical and effective, enabling the application of language models to
address specific industry or domain requirements.

###  IX-D Ethical Considerations and Bias Mitigation:

Addressing ethical considerations, including bias mitigation and fairness, was
a crucial aspect of prompt engineering during this period. As large language
models became more accessible, prompt engineering techniques placed increased
emphasis on responsible AI usage. Researchers developed strategies to de-bias
prompts, [23], define fairness-related control codes, [24], and craft reward
functions that encouraged unbiased and equitable language generation. The aim
was to ensure prompt engineering adhered to ethical guidelines, promoting
responsible and fair AI systems.

###  IX-E Interpretability and Explainability:

Furthermore, interpretability and explainability became important areas of
focus in prompt engineering. Researchers explored methods to visualize
attention weights and understand how language models made decisions based on
their input prompts, [25]. Explainable prompt engineering allowed users to
gain insights into model behavior, enhancing transparency and trust in AI
systems.

###  IX-F Democratization of AI and Low-Resource Language Support:

Moreover, the democratization of AI technology during 2020 and 2021 enabled
wider access to powerful language models and prompt engineering techniques,
[26, 27]. Developers could integrate prompt engineering into various
applications and services, making AI-driven language generation more
accessible to diverse industries and use cases. Additionally, efforts were
made to extend prompt engineering to support low-resource languages, ensuring
that language models were inclusive and representative of a wide range of
linguistic diversity.

In conclusion, the years 2020 and 2021 marked a transformative era in prompt
engineering, with the rise of massive language models and innovations in
prompt design. The integration of prompt engineering with large-scale language
models democratized AI technology and made it more inclusive. Ethical
considerations and interpretability played a significant role in ensuring
responsible AI usage. Prompt engineering techniques enabled researchers and
developers to create AI systems that were more contextually aware,
controllable, and adaptable, paving the way for a wide range of applications
and advancements in natural language processing.

##  X 2022-Current: Beyond Language Generation

Prompt #9: (a) Can you now write a section on 2022 and 2023 on advanced prompt
techniques? (b) can you write the section on 2022 and 2023 on advanced prompt
techniques in paragraphs instead of bullet points?

The years 2022 and 2023 witnessed remarkable advancements in prompt
engineering, pushing the boundaries of language models and extending their
applications beyond conventional language generation tasks. Prompt techniques
evolved to address complex challenges, such as multimodal inputs, multi-turn
conversations, and domain-specific language understanding. Researchers
explored techniques that augmented prompt engineering with additional context
and domain knowledge, making AI systems more contextually aware, interactive,
and versatile. Here’s an overview of the key advanced prompt techniques that
shaped the field during this period:

###  X-A Multimodal Prompting and Integration:

One of the most notable advancements during this period was the integration of
multimodal prompting. Prompt engineering expanded to include various input
modalities, combining textual prompts with visual, auditory, or other sensory
information. By seamlessly incorporating visual elements like images or videos
into textual prompts, AI systems gained the ability to process and generate
content from diverse sources. This paved the way for applications in image
captioning, visual question answering, and interactive chatbots, enabling AI
systems to understand and generate responses in a more holistic and
comprehensive manner.

###  X-B Multi-Turn Conversational Prompting:

In parallel, the focus of prompt engineering expanded from single-turn
language generation to multi-turn conversational prompting. Techniques were
developed to maintain and utilize context across multiple interactions,
allowing AI systems to engage in more coherent and interactive conversations
with users. Multi-turn conversational prompting greatly enhanced chatbot
capabilities, making AI assistants more responsive and proficient in handling
complex user queries and extended discussions..

###  X-C Domain-Specific Knowledge Integration:

Moreover, the integration of domain-specific knowledge became a significant
aspect of prompt engineering in 2022 and 2023. Researchers explored methods to
incorporate external knowledge bases or leverage pre-existing domain-specific
models to enhance the language understanding and generation process. By
tapping into specialized knowledge, AI systems demonstrated improved
performance in domain-specific applications such as medicine, law, finance,
and more, effectively bridging the gap between AI and domain expertise.

###  X-D Reinforcement Learning with Human Feedback:

Reinforcement learning techniques continued to play a crucial role in prompt
engineering, especially in refining AI systems’ controllability and
adaptability. Researchers explored strategies to integrate human feedback as
reward signals for fine-tuning models. Human-in-the-loop prompt engineering
allowed for more effective and interactive model refinement, reducing the need
for extensive manual annotation and providing users with personalized and
tailored responses.

###  X-E Contextual Intent Prediction:

Contextual intent prediction emerged as another significant area of focus.
Prompt engineering techniques in 2022 and 2023 aimed to identify user queries’
underlying intent to generate contextually relevant and accurate responses. By
predicting user intent from input prompts, AI systems could better align with
user goals and needs, offering more personalized and targeted language
generation.

###  X-F Personalization and User Modeling:

Furthermore, prompt engineering techniques in 2022 and 2023 saw an increased
emphasis on personalization and user modeling. AI systems were designed to
learn from past interactions and form user-specific models, which informed
prompt engineering decisions to generate personalized content tailored to
individual users. Personalization strategies enhanced user engagement and
satisfaction, allowing AI systems to adapt responses based on user preferences
and behavior.

In conclusion, the years 2022 and 2023 brought forth a series of
groundbreaking advancements in prompt engineering. The integration of
multimodal inputs, multi-turn conversational prompting, domain-specific
knowledge, reinforcement learning, contextual intent prediction, and user
modeling collectively transformed AI systems into contextually intelligent and
versatile language generators. As prompt engineering continued to evolve,
researchers and developers explored novel applications and use cases, paving
the way for AI systems that are highly adaptable, interactive, and capable of
serving diverse user needs across various domains.

##  XI CONCLUSIONS

In conclusion, the history of prompt engineering in natural language
processing has been a journey of groundbreaking advancements and innovation.
From the early language models and information retrieval systems to the
transformative introduction of attention mechanisms, reinforcement learning,
and advanced language models, prompt engineering has continually evolved to
enhance language understanding, generation, and controllability.

The attention mechanism, introduced in 2015, revolutionized language modeling
by allowing models to focus on relevant information and establish contextual
relationships, leading to improved performance across various NLP tasks.
Reinforcement learning techniques in 2017 further enhanced controllability,
enabling prompt engineers to shape language models with rewarding strategies
and address biases.

Advancements in 2018 and 2019 introduced fine-tuning strategies, control
codes, and template-based generation, refining controllability and context-
awareness in language models. Research focused on fairness, human-AI
collaboration, and adaptation to low-resource languages, marking strides
towards responsible and inclusive AI systems.

In 2020 and 2021, contextual prompting and transfer learning were prominent,
making language models more interactive and adaptable. These developments
extended to 2022 and 2023, introducing advanced techniques like unsupervised
pre-training and novel reward shaping.

The history of prompt engineering showcases the collaborative efforts of
researchers and practitioners driving the field’s evolution. With ethical
considerations, prompt engineering promises to revolutionize AI applications,
fostering responsible human-AI collaboration and empowering language models to
align closely with human intentions.

In conclusion, prompt engineering continues to shape the landscape of NLP,
opening new frontiers for intelligent and context-aware language models. The
journey is ongoing, and with continuous research and responsible development,
prompt engineering will propel the field towards more versatile,
interpretable, and human-centric AI systems.

## Acknowledgment

Authors extensively used ChatGPT for content generation.

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# A novel approach for deriving the stable boundary layer height and
eddy viscosity profiles from the Ekman equations

Sukanta Basu<EMAIL_ADDRESS>Faculty of Civil Engineering and
Geosciences, Delft University of Technology, Delft, the Netherlands  Albert A.
M. Holtslag<EMAIL_ADDRESS>Meteorology and Air Quality, Wageningen
University, Wageningen, the Netherlands

###### Abstract

In this study, we utilize a novel approach to solve the Ekman equations for
eddy viscosity profiles in the stable boundary layer. By doing so, a well-
known expression for the stable boundary layer height by Zilitinkevich
(Boundary-Layer Meteorology, 1972, Vol. 3, 141–145) is rediscovered.

Boundary layer height; Eddy viscosity; Ekman equations; K profile; Stable
boundary layer

##  I Introduction

Sergej Zilitinkevich was one of the giants of atmospheric physics who carried
the field of boundary-layer meteorology (BLM) on his shoulder for more than
half a century. We, representing the BLM community at large, are indebted
forever for his ingenious efforts and lifelong dedication in advancing our
field. In the arena of stable boundary layers (SBLs), Zilitinkevich made
numerous ground-breaking contributions. As a matter of fact, it would be
difficult to find any contemporary article on SBLs which does not make at
least one reference to an original publication of Zilitinkevich. The present
paper is also following the same tradition.

In this study, we utilize the Ekman equations to analytically derive a stable
boundary layer height formulation which was originally proposed by
Zilitinkevich Zilitinkevich (1972) based on scaling arguments. During the
process, we also derive equations for the vertical profiles of eddy viscosity.
To the best of our knowledge, it is the first time that estimates are given
for the eddy viscosity profiles directly from the Ekman equations.

##  II Formulation of SBL Height by Zilitinkevich (1972)

Using boundary layer scaling arguments, Zilitinkevich Zilitinkevich (1972)
proposed that the height ($h$) of stationary SBLs can be written as:

$h=\gamma\sqrt{\frac{u_{*0}L}{f_{cor}}}=C_{h}u_{*0}^{2}\left|f_{cor}B_{s}\right|^{-1/2}.$
(1)

Here the surface friction velocity and surface Obukhov length are denoted by
$u_{*0}$ and $L$, respectively; $B_{s}$ is the buoyancy flux. The Coriolis
term is represented by $f_{cor}$. The proportionality constants $\gamma$ and
$C_{h}$ are related as follows:

$C_{h}=\frac{\gamma}{\sqrt{\kappa}}\approx 1.58\gamma,$ (2)

where $\kappa$ is the von Kármán constant assumed to be equal to 0.4.
Zilitinkevich Zilitinkevich (1972) assumed $C_{h}$ to be order of one. In an
analytical study, Businger and Arya Businger and Arya (1974) estimated
$\gamma$ to be equal to 0.72. A much lower value of $\gamma\approx 0.4$ was
estimated by Brost and Wyngaard Brost and Wyngaard (1978) using a second-order
closure model. Garratt Garratt (1982) who analyzed observational data from
several field campaigns also found $\gamma\approx 0.4$. In his local-scaling
paper, Nieuwstadt Nieuwstadt (1984) found $\gamma\approx 0.35$ to be
consistent with other equations. Zilitinkevich Zilitinkevich (1989) summarized
$C_{h}$ from several studies and found it to vary within the range of
0.55–1.58. A unique aspect of the present study is that we analytically derive
$\gamma$ (and $C_{h}$) with limited assumptions.

We would like to point out that Eq. 1 predicts physically unrealistic boundary
layer heights for two situations: (i) close to the equator (i.e., $f_{cor}\to
0$); and (ii) for near-neutral conditions (i.e., $L\to\infty$). To circumvent
the second issue, a few interpolation approaches have been proposed in the
literature (e.g., Nieuwstadt, 1981; Zilitinkevich, 1989).

##  III Momentum and Sensible Heat Flux Profiles

In SBL flows, the profiles of friction velocity and sensible heat flux are
often expressed as follows:

$u_{*L}^{2}=\left(\overline{uw}_{L}^{2}+\overline{vw}_{L}^{2}\right)^{1/2}=u_{*0}^{2}\left(1-\frac{z}{h}\right)^{\alpha}=u_{*0}^{2}f_{m},$
(3a)
$\overline{w\theta}_{L}=\overline{w\theta}_{0}\left(1-\frac{z}{h}\right)^{\beta}=\overline{w\theta}_{0}f_{h}.$
(3b)

Here the friction velocity, momentum flux components, and sensible heat flux
at height $z$ are denoted by $u_{*L}$, $\overline{uw}_{L}$,
$\overline{vw}_{L}$, and $\overline{w\theta}_{L}$, respectively. The subscript
‘0’ is used to demarcate the corresponding surface values. These equations
imply that the turbulent fluxes are maximum near the surface and they
monotonically decrease to zero at the top of the boundary layer.

The exponents $\alpha$ and $\beta$ in Eqs. 3a and 3b are not universal
constants. By utilizing his local-scaling hypothesis, Nieuwstadt Nieuwstadt
(1984) suggested $\alpha=1.5$. He also proved that $\beta$ should be equal to
1 under the assumptions of horizontal homogeneity and stationarity. In
contrast, based on observational data from the well-known Minnesota field
campaign (Caughey  _et al._ , 1979), Sorbjan Sorbjan (1986) estimated
$\alpha=2$, and $\beta=3$ for evolving SBLs. In another observational study,
Lenschow et al. Lenschow  _et al._ (1988) considered the additional effects of
radiational cooling and found the optimal $\alpha$ and $\beta$ to be equal to
1.75 and 1.5, respectively.

Using the definition of Obukhov length (Stull, 1988) in conjunction with Eqs.
3a and 3b, we get:

$\Lambda=L\frac{f_{m}^{3/2}}{f_{h}},$ (4)

where $\Lambda$ is the so-called local Obukhov length at height $z$.

The first and second derivatives of $f_{m}$ function in Eq. 3a can be written
as:

$f_{m}^{\prime}=-\frac{\alpha}{h}\left(1-\frac{z}{h}\right)^{\alpha-1},$ (5a)
$f_{m}^{\prime\prime}=\frac{\alpha\left(\alpha-1\right)}{h^{2}}\left(1-\frac{z}{h}\right)^{\alpha-2}.$
(5b)

We will make use of these derivatives in a later section.

##  IV Eddy Viscosity Profile in the Ekman Layer

According to the K-theory, based on the celebrated hypothesis of Boussinesq in
1877, turbulent fluxes can be approximated as products of the eddy exchange
coefficients and the mean gradients (Lumley and Panofsky, 1964).

$\overline{uw}_{L}=-K_{M}\frac{\partial U}{\partial z},$ (6a)
$\overline{vw}_{L}=-K_{M}\frac{\partial V}{\partial z}.$ (6b)

Here $K_{M}$ is the so-called eddy viscosity coefficient. Based on these
equations, we can write:

$u_{*L}^{2}=\left(\overline{uw}_{L}^{2}+\overline{vw}_{L}^{2}\right)^{1/2}=K_{M}S,$
(7)

where $S$ is the magnitude of wind speed shear.

For boundary layer flows over homogeneous and flat terrain, under steady-state
conditions, the averaged equations of motions can be simplified as follows
(Zilitinkevich  _et al._ , 1967; Brown, 1974):

$f_{cor}\left(V-V_{g}\right)=\frac{\partial\left(\overline{uw}_{L}\right)}{\partial
z},$ (8a)
$f_{cor}\left(U-U_{g}\right)=-\frac{\partial\left(\overline{vw}_{L}\right)}{\partial
z}.$ (8b)

The velocity components in $x$ and $y$ directions are represented as $U$ and
$V$, respectively. Similarly, the geostrophic velocity components are denoted
by $U_{g}$ and $V_{g}$. In the literature, these equations are commonly known
as the Ekman layer equations (Brown, 1974; Nieuwstadt, 1983).

In a landmark paper, Ekman Ekman (1905) first analytically solved Eqs. 8a and
8b with the assumption of constant eddy viscosity ($K_{M}$) in conjunction
with appropriate boundary conditions. Over the years, a few more closed form
analytical solutions of the Ekman equations have been reported in the
literature (e.g., Wippermann, 1973; Brown, 1974; Nieuwstadt, 1983; Grisogono,
1995; Parmhed  _et al._ , 2005). In all these papers, simplified profiles of
$K_{M}$ were always assumed. In this paper, we take a radically different
approach. We only assume that Eq. 3a is valid and then deduce $K_{M}$ profile
from the Ekman equations as shown below.

For barotropic condition, the Ekman equations can be re-written as:

$f_{cor}\frac{\partial V}{\partial
z}=\frac{\partial^{2}\left(\overline{uw}_{L}\right)}{\partial z^{2}},$ (9a)
$f_{cor}\frac{\partial U}{\partial
z}=-\frac{\partial^{2}\left(\overline{vw}_{L}\right)}{\partial z^{2}}.$ (9b)

Next, by utilizing Eqs. 6a and 6b, these equations are transformed as follows:

$-f_{cor}\frac{\overline{vw}_{L}}{K_{M}}=\frac{\partial^{2}\left(\overline{uw}_{L}\right)}{\partial
z^{2}},$ (10a)
$-f_{cor}\frac{\overline{uw}_{L}}{K_{M}}=-\frac{\partial^{2}\left(\overline{vw}_{L}\right)}{\partial
z^{2}}.$ (10b)

Dividing Eq. 10a by Eq. 10b and rearranging we arrive at:

$\overline{uw}_{L}\frac{\partial^{2}\left(\overline{uw}_{L}\right)}{\partial
z^{2}}+\overline{vw}_{L}\frac{\partial^{2}\left(\overline{vw}_{L}\right)}{\partial
z^{2}}=0.$ (11)

The momentum flux components can be decomposed in terms of local friction
velocity as follows:

$\overline{uw}_{L}=-u_{*L}^{2}\cos\left(\delta\right),$ (12a)
$\overline{vw}_{L}=u_{*L}^{2}\sin\left(\delta\right).$ (12b)

Here $\delta$ is the angle between the flux vector and the $x$-axis. By
plugging Eq. 3a in Eqs. 12a and 12b, we get:

$\overline{uw}_{L}=-u_{*0}^{2}f_{m}\cos\left(\delta\right),$ (13a)
$\overline{vw}_{L}=u_{*0}^{2}f_{m}\sin\left(\delta\right).$ (13b)

These equations can be differentiated as follows:

$\frac{\partial\left(\overline{uw}_{L}\right)}{\partial
z}=-u_{*0}^{2}\left[f^{\prime}_{m}\cos(\delta)-f_{m}\delta^{\prime}\sin(\delta)\right],$
(14a) $\frac{\partial\left(\overline{vw}_{L}\right)}{\partial
z}=u_{*0}^{2}\left[f^{\prime}_{m}\sin(\delta)+f_{m}\delta^{\prime}\cos(\delta)\right].$
(14b)

After differentiating one more time we arrive at:

$\frac{\partial^{2}\left(\overline{uw}_{L}\right)}{\partial
z^{2}}=-u_{*0}^{2}\left[f^{\prime\prime}_{m}\cos(\delta)-2f^{\prime}_{m}\delta^{\prime}\sin(\delta)-f_{m}\delta^{\prime\prime}\sin(\delta)-f_{m}\delta^{\prime
2}\cos(\delta)\right],$ (15a)
$\frac{\partial^{2}\left(\overline{vw}_{L}\right)}{\partial
z^{2}}=u_{*0}^{2}\left[f^{\prime\prime}_{m}\sin(\delta)+2f^{\prime}_{m}\delta^{\prime}\cos(\delta)+f_{m}\delta^{\prime\prime}\cos(\delta)-f_{m}\delta^{\prime
2}\sin(\delta)\right].$ (15b)

By combining Eqs. 11, 13a, 13b, 15a, 15b, and simplifying we get:

$f_{m}f^{\prime\prime}_{m}\cos^{2}(\delta)-2f_{m}f^{\prime}_{m}\delta^{\prime}\sin(\delta)\cos(\delta)-f_{m}^{2}\delta^{\prime\prime}\sin(\delta)\cos(\delta)-f_{m}^{2}\delta^{\prime
2}\cos^{2}(\delta)+\\\
f_{m}f^{\prime\prime}_{m}\sin^{2}(\delta)+2f_{m}f^{\prime}_{m}\delta^{\prime}\sin(\delta)\cos(\delta)+f_{m}^{2}\delta^{\prime\prime}\sin(\delta)\cos(\delta)-f_{m}^{2}\delta^{\prime
2}\sin^{2}(\delta)=0.$ (16)

By invoking the Pythagorean trigonometric identity, we can further simplify
this equation to:

$f_{m}f^{\prime\prime}_{m}-f^{2}_{m}\delta^{\prime 2}=0.$ (17)

Thus,

$\delta^{\prime}=\sqrt{\frac{f^{\prime\prime}_{m}}{f_{m}}}.$ (18)

Substituting $f_{m}$ and $f_{m}^{\prime\prime}$ from Eqs. 3a and 5b in 18, we
find:

$\delta^{\prime}=\left(\frac{\sqrt{\alpha\left(\alpha-1\right)}}{h}\right)\left(1-\frac{z}{h}\right)^{-1}.$
(19)

The second derivative of $\delta$ is:

$\delta^{\prime\prime}=\frac{\sqrt{\alpha\left(\alpha-1\right)}}{h^{2}}\left(1-\frac{z}{h}\right)^{-2}.$
(20)

Please note that these derivatives have real values if $\alpha$ is greater
than one.

Now, we can directly estimate the $K_{M}$ profile from Eqs. 10a, 13b, 15a, 19,
and 20:

$\displaystyle K_{M}$
$\displaystyle=-\frac{f_{cor}\overline{vw}_{L}}{\partial^{2}(\overline{uw}_{L})/\partial
z^{2}}$ (21a)
$\displaystyle=+\frac{f_{cor}f_{m}\sin(\delta)}{f^{\prime\prime}_{m}\cos(\delta)-2f^{\prime}_{m}\delta^{\prime}\sin(\delta)-f_{m}\delta^{\prime\prime}\sin(\delta)-f_{m}\delta^{\prime
2}\cos(\delta)}$ (21b)
$\displaystyle=\frac{f_{cor}h^{2}}{\left(2\alpha-1\right)\sqrt{\alpha\left(\alpha-1\right)}}\left(1-\frac{z}{h}\right)^{2}.$
(21c)

We would like to emphasize that this formulation of $K_{M}$ is derived
directly from the Ekman equations with very limited assumptions. To the best
of our knowledge, a similar formulation and derivation have not been reported
in the literature.

Similar to other Ekman layer findings, Eq. 21 is only valid in the outer
layer. It does not represent surface layer conditions. In the literature,
various patching and asymptotic matching approaches (e.g., Taylor, 1915;
Blackadar and Tennekes, 1968; Brown, 1974; Zilitinkevich, 1975) have been
proposed to combine outer layer and inner layer (i.e., surface layer)
solutions. A nice overview was given by Hess and Garratt (2002). In Sects. VI
and VII, we utilize an unorthodox strategy.

##  V Conventional K-profile Approach

One of the most widely used first-order formulation for $K_{M}$ is the
K-profile approach (Stensrud, 2007). O’Brien O’Brien (1970) was one of the
first researchers to propose a K-profile which portrays desirable surface
layer behavior, attains a maximum value within the planetary boundary layer
(PBL), and decreases to a background diffusion level above the PBL. Based on a
second-order closure model, Brost and Wyngaard Brost and Wyngaard (1978)
proposed a different K-profile formulation for stably stratified flows:

$K_{M}=\frac{\left(\kappa
zu_{*0}\right)}{\phi_{M}}\left(1-\frac{z}{h}\right)^{p}.$ (22)

Here $\phi_{M}$ is a type of non-dimensional velocity gradient, defined later.
The exponent $p$ is a-priori not known. For neutrally stratified flows in the
surface layer, Eq. 22 reduces to $K_{M}=\kappa zu_{*}$; this equation is in
complete agreement with the well-established logarithmic law of the wall.
Furthermore, for stably stratified surface layers, one can deduce a stability-
corrected logarithmic law of the wall (e.g., Businger  _et al._ , 1971) from
Eq. 22.

The K-profile approach was modified by several researchers Holtslag and Moeng
(1991); Holtslag and Boville (1993); Hong and Pan (1996); Hong  _et al._
(2006); Noh  _et al._ (2003); Troen and Mahrt (1986) for its application in
the unstable regime. They included a counter-gradient term to include the
effects of large-scale eddies on local fluxes. They also considered
entrainment fluxes at the top of the PBL.

In the context of numerical stability, the K-profile approach is quite robust
(Beljaars, 1992). Thus, it is not surprising that it is widely used in
numerical weather prediction models. As a matter of fact, the default PBL
scheme (called the YSU scheme) of the popular Weather Research and Forecasting
(WRF) model uses the K-profile approach for both unstable and stable
conditions.

In spite of its wide usage, Eq. 22 suffers from two limitations. First, there
is uncertainty in the value of the exponent $p$. Brost and Wyngaard Brost and
Wyngaard (1978) found $p$ to be equal to 1.5 based on their simulations. On
the other hand, based on field campaign data from Minnesota, Sorbjan Sorbjan
(1989) found $p=1$. In the absence of reliable field observations Troen and
Mahrt Troen and Mahrt (1986) used an integer value of $p=2$. The local scaling
hypothesis by Nieuwstadt Nieuwstadt (1984) also leads to $p=2$. In the present
study, we estimate $p$ analytically.

The other limitation is related to the parameterization of $\phi_{M}$. The
K-profile formulation uses the following normalized velocity gradient:

$\phi_{M}=\left(\frac{\kappa zS}{u_{*0}}\right).$ (23a) Please note that in
this equation surface friction velocity ($u_{*0}$) is used. However, in
contrast to well-known surface layer formulations, $z$ is not limited to the
depth of the surface layer. Instead, $z$ ranges from the surface to the top of
the boundary layer. $\phi_{M}$ is commonly parameterized as Brost and Wyngaard
(1978):
$\phi_{M}=1+c\left(\frac{z}{L}\right)=1+c\left(\frac{z}{h}\right)\left(\frac{h}{L}\right),$
(23b)

where $c$ is a constant and often assumed equal to 5. In the surface layer,
for $z/L<1$, numerous studies documented the validity of this equation.
However, its applicability for the outer layer (i.e., above surface layer) is
questionable. Furthermore, this equation (incorrectly) implies that the
logarithmic law of the wall applies to the entire boundary layer for neutral
condition (i.e., $h/L=0$).

##  VI Alternative K-profile Approaches

In this section, we derive two competing K-profile formulations. Both the
formulations are applicable for the entire SBL (i.e., including the surface
layer and the outer layer).

###  VI.1 Option 1

Multiplying both sides of Eq. 7 by $\left(\kappa z/u_{*0}\right)$, we get:

$\left(\frac{\kappa z}{u_{*0}}\right)u_{*L}^{2}=K_{M}\left(\frac{\kappa
zS}{u_{*0}}\right),$ (24a) By using Eq. 3a, we deduce: $\left(\kappa
zu_{*0}\right)f_{m}=K_{M}\phi_{M}.$ (24b) Thus, $K_{M}=\frac{\left(\kappa
zu_{*0}\right)f_{m}}{\phi_{M}}=\frac{\left(\kappa
zu_{*0}\right)}{\phi_{M}}\left(1-\frac{z}{h}\right)^{\alpha}.$ (24c)

This equation is identical to the one proposed by Brost and Wyngaard Brost and
Wyngaard (1978) based on second-order modeling. Except, in case of Eq. 24c,
the exponent $\alpha$ is the same as in Eq. 3a.

###  VI.2 Option 2

An alternate expression for $K_{M}$ profile can be found by multiplying Eq. 7
by $(\kappa z/u_{*L})$:

$\left(\frac{\kappa z}{u_{*L}}\right)u_{*L}^{2}=K_{M}\left(\frac{\kappa
zS}{u_{*L}}\right),$ (25a) or, $\left(\kappa zu_{*L}\right)=K_{M}\phi_{ML}.$
(25b) By using Eq. 3a, we get: $\left(\kappa
zu_{*0}\right)f_{m}^{1/2}=K_{M}\phi_{ML},$ (25c) Hence,
$K_{M}=\frac{\left(\kappa
zu_{*0}\right)f_{m}^{1/2}}{\phi_{ML}}=\frac{\left(\kappa
zu_{*0}\right)}{\phi_{ML}}\left(1-\frac{z}{h}\right)^{\alpha/2}.$ (25d)

In these equations, $\phi_{ML}(=\kappa zS/u_{*L})$ is a local non-dimensional
velocity gradient as it utilizes local friction velocity ($u_{*L}$) from
height $z$. It is straightforward to show that:

$\phi_{M}=\phi_{ML}f_{m}^{1/2}.$ (26)

Equation 26 implies that $\phi_{M}$ decreases more strongly with height than
$\phi_{ML}$. Several past simulation studies (e.g., Basu and Porté-Agel, 2006;
Zilitinkevich and Esau, 2007; van de Wiel  _et al._ , 2008) found the
following parameterization for $\phi_{ML}$:

$\phi_{ML}=1+c_{L}\left(\frac{z}{\Lambda}\right).$ (27)

In those studies, $c_{L}$ was found to be between 3 and 5. By regression
analysis of field observations, Mahrt and Vickers Mahrt and Vickers (2003)
found $c_{L}$ = 3.7. Here, we assume $c_{L}=4$.

##  VII Matching of $K_{M}$ Profiles in the Outer Layer

Thus far, we have derived 3 different K-profiles. Note that Eq. 21 is only
valid in the outer layer. In contrast, Eqs. 24c and 25d are valid in the
entire SBL. In the following sub-sections, we match these $K_{M}$ profiles for
the outer layer when $z/L\gg 1$ and $z/\Lambda\gg 1$.

###  VII.1 Option 1

In the outer layer, for $z/L\gg 1$, Eq. 24c simplifies to:

$\displaystyle K_{M}$ $\displaystyle=\frac{\left(\kappa
zu_{*0}\right)f_{m}}{c\left(\frac{z}{L}\right)},$ (28a)
$\displaystyle=\left(\frac{\kappa}{c}\right)\left(u_{*0}L\right)f_{m},$ (28b)
$\displaystyle=\left(\frac{\kappa}{c}\right)\left(u_{*0}L\right)\left(1-\frac{z}{h}\right)^{\alpha}.$
(28c)

If we assume $\alpha=2$, then we get:

$K_{M}=\left(\frac{\kappa}{c}\right)\left(u_{*0}L\right)\left(1-\frac{z}{h}\right)^{2}.$
(29)

Similarly, by plugging in $\alpha=2$ in Eq. 21, we get:

$K_{M}=\frac{f_{cor}h^{2}}{3\sqrt{2}}\left(1-\frac{z}{h}\right)^{2}.$ (30)

By directly matching Eq. 29 with Eq. 30 we arrive at:

$\frac{f_{cor}h^{2}}{3\sqrt{2}}=\left(\frac{\kappa}{c}\right)\left(u_{*0}L\right).$
(31)

Further simplification leads to the SBL height parameterization of
Zilitinkevich Zilitinkevich (1972):

$h=\gamma_{1}\sqrt{\frac{u_{*0}L}{f_{cor}}},$ (32a) where
$\gamma_{1}=\sqrt{3\sqrt{2}\left(\frac{\kappa}{c}\right)}.$ (32b)

For $c=5$, we have: $\gamma_{1}=0.583$.

###  VII.2 Option 2

Similar to option 1, in the outer layer, for $z/\Lambda\gg 1$, Eq. 25d
simplifies to:

$\displaystyle K_{M}$ $\displaystyle=\frac{\left(\kappa
zu_{*0}\right)f_{m}^{1/2}}{c_{L}\left(\frac{z}{\Lambda}\right)},$ (33a)
$\displaystyle=\left(\frac{\kappa}{c_{L}}\right)\left(u_{*0}\Lambda\right)f_{m}^{1/2},$
(33b)
$\displaystyle=\left(\frac{\kappa}{c_{L}}\right)\left(\frac{u_{*0}Lf_{m}^{2}}{f_{h}}\right).$
(33c)

In this derivation, we used Eq. 4 in the conversion of $\Lambda$ to $L$. If we
assume $\alpha=3/2$ and $\beta=1$, we get:

$K_{M}=\left(\frac{\kappa}{c_{L}}\right)\left(u_{*0}L\right)\left(1-\frac{z}{h}\right)^{2}.$
(34)

Comparing this equation with Eq. 21, we arrive at:

$h=\gamma_{2}\sqrt{\frac{u_{*0}L}{f_{cor}}},$ (35a) where
$\gamma_{2}=\sqrt{\sqrt{3}\left(\frac{\kappa}{c_{L}}\right)}.$ (35b)

If $c_{L}=4$, we get $\gamma_{2}=0.416$.

Interestingly, both the options 1 and 2 lead to the SBL height
parameterization of Zilitinkevich Zilitinkevich (1972). The estimated
proportionality constants, $\gamma_{1}$ and $\gamma_{2}$, fall within the
range of previous observation-based and simulation-based empirical values.
Furthermore, based on the literature, both $\alpha=3/2$ and $\alpha=2$ are
plausible in SBLs. However, using $\phi_{M}=1+5z/L$ for the entire SBL, as
commonly done in practice, does not seem physically meaningful and is in
contrast with observations (e.g., Holtslag, 1984, among many others). From
this perspective, option 2 seems to be a better option.

##  VIII Conclusion

Fifty years ago, Zilitinkevich Zilitinkevich (1972) proposed a formulation for
SBL height by using boundary layer scaling arguments. In this study, we derive
the same formulation from an analytical approach involving the Ekman layer
equations. In addition, we provide novel derivations for eddy viscosity
profiles in the SBL. Our approach makes use of the following assumptions: (i)
turbulent fluxes decrease monotonically with height and go to zero at the top
of the boundary layer; (ii) the K theory is applicable for stably stratified
conditions; and (iii) under steady-state condition, for flows over homogeneous
and flat terrain, geostrophic balance holds in the boundary layer. None of
these assumptions are unorthodox. In our future work, we hope to extend our
analytical approach to derive geostrophic drag laws for SBLs.

###### Acknowledgements.

The first author is indebted to Pouriya Alinaghi, Branko Kosović, and
Arquímedes Ruiz-Columbié for useful discussions and their assistance in cross-
checking the analytical derivations. The second author acknowledges the many
discussions and interactions he had with Sergej Zilitinkevich over the years.

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